1.
Subset
–
In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
2.
Simply connected space
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If a space is not simply-connected, it is convenient to measure the extent to which it fails to be simply-connected, this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space, if there are no holes, the group is trivial — equivalently. Informally, an object in our space is simply-connected if it consists of one piece. For example, neither a doughnut nor a cup is simply connected. In two dimensions, a circle is not simply-connected, but a disk and a line are, spaces that are connected but not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected. To illustrate the notion of connectedness, suppose we are considering an object in three dimensions, for example, an object in the shape of a box, a doughnut. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a piece of string and trails it through the water inside the aquarium, in whatever way he pleases. Now the loop begins to contract on itself, getting smaller and smaller, if the loop can always shrink all the way to a point, then the aquariums interior is simply connected. If sometimes the loop gets caught — for example, around the hole in the doughnut — then the object is not simply-connected. Notice that the only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point. The stronger condition, that the object has no holes of any dimension, is called contractibility, intuitively, this means that p can be continuously deformed to get q while keeping the endpoints fixed. Hence the term simply connected, for any two points in X, there is one and essentially only one path connecting them. A third way to express the same, X is simply-connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i. e. consists only of the identity element. Yet another formulation is used in complex analysis, an open subset X of C is simply-connected if. It might also be worth pointing out that a relaxation of the requirement that X be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a set has connected extended complement exactly when each of its connected components are simply-connected
3.
Genus (mathematics)
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It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ =2 − 2g for closed surfaces, for surfaces with b boundary components, the equation reads χ =2 − 2g − b. In laymans terms, its the number of holes an object has, a doughnut, or torus, has 1 such hole. The green surface pictured above has 2 holes of the relevant sort, for instance, The sphere S2 and a disc both have genus zero. A torus has genus one, as does the surface of a mug with a handle. This is the source of the joke that a topologist is someone who cant tell his donut from his coffee mug, an explicit construction of surfaces of genus g is given in the article on the fundamental polygon. Genus of orientable surfaces In simpler terms, the value of an orientable surfaces genus is equal to the number of holes it has. The non-orientable genus, demigenus, or Euler genus of a connected, alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ =2 − k, where k is the non-orientable genus. For instance, A projective plane has non-orientable genus one, a Klein bottle has non-orientable genus two. The genus of a knot K is defined as the genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary the boundary being the knot, the genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary. The genus of a 3-dimensional handlebody is an integer representing the number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it, for instance, A ball has genus zero. A solid torus D2 × S1 has genus one, the genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles. Thus, a graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps. The Euler genus is the integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles. In topological graph theory there are definitions of the genus of a group
4.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
5.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
6.
Topological space
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Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion, the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. The utility of the notion of a topology is shown by the fact there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application, the most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so this is given first. Note, A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy and this axiomatization is due to Felix Hausdorff. Let X be a set, the elements of X are usually called points, let N be a function assigning to each x in X a non-empty collection N of subsets of X. The elements of N will be called neighbourhoods of x with respect to N, the function N is called a neighbourhood topology if the axioms below are satisfied, and then X with N is called a topological space. If N is a neighbourhood of x, then x ∈ N, in other words, each point belongs to every one of its neighbourhoods. If N is a subset of X and includes a neighbourhood of x, I. e. every superset of a neighbourhood of a point x in X is again a neighbourhood of x. The intersection of two neighbourhoods of x is a neighbourhood of x, any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M. The first three axioms for neighbourhoods have a clear meaning, the fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of X. A standard example of such a system of neighbourhoods is for the real line R, given such a structure, we can define a subset U of X to be open if U is a neighbourhood of all points in U. A topological space is a pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms, The empty set. Any union of members of τ still belongs to τ, the intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X, given X =, the collection τ = of only the two subsets of X required by the axioms forms a topology of X, the trivial topology. Given X =, the collection τ = of six subsets of X forms another topology of X, given X = and the collection τ = P, is a topological space. τ is called the discrete topology, using de Morgans laws, the above axioms defining open sets become axioms defining closed sets, The empty set and X are closed. The intersection of any collection of closed sets is also closed, the union of any finite number of closed sets is also closed
7.
Union (set theory)
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In set theory, the union of a collection of sets is the set of all elements in the collection. It is one of the operations through which sets can be combined and related to each other. For explanation of the used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, for example, if A = and B = then A ∪ B =. Sets cannot have duplicate elements, so the union of the sets and is, multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. Binary union is an operation, that is, A ∪ = ∪ C. The operations can be performed in any order, and the parentheses may be omitted without ambiguity, similarly, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union and that is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction, since sets with unions and intersections form a Boolean algebra, intersection distributes over union A ∩ = ∪ and union distributes over intersection A ∪ = ∩. One can take the union of several sets simultaneously, for example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. In mathematics a finite union means any union carried out on a number of sets. The most general notion is the union of a collection of sets. If M is a set whose elements are themselves sets, then x is an element of the union of M if, in symbols, x ∈ ⋃ M ⟺ ∃ A ∈ M, x ∈ A. This idea subsumes the preceding sections, in that A ∪ B ∪ C is the union of the collection, also, if M is the empty collection, then the union of M is the empty set. The notation for the concept can vary considerably. For a finite union of sets S1, S2, S3, …, S n one often writes S1 ∪ S2 ∪ S3 ∪ ⋯ ∪ S n or ⋃ i =1 n S i. In the case that the index set I is the set of natural numbers, whenever the symbol ∪ is placed before other symbols instead of between them, it is of a larger size
8.
Disjoint set
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In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set, for example, and are disjoint sets, while and are not. This definition of disjoint sets can be extended to any family of sets, a family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets is pairwise disjoint, two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two sets whose intersection is a finite set may be said to be almost disjoint. In topology, there are notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods, similarly, in a metric space, positively separated sets are sets separated by a nonzero distance. Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections, two sets A and B are disjoint if and only if their intersection A ∩ B is the empty set. It follows from definition that every set is disjoint from the empty set. A family F of sets is pairwise disjoint if, for two sets in the family, their intersection is empty. If the family more than one set, this implies that the intersection of the whole family is also empty. However, a family of one set is pairwise disjoint, regardless of whether that set is empty. Additionally, a family of sets may have an empty intersection without being pairwise disjoint, for instance, the three sets have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection, also the empty family of sets is pairwise disjoint. A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the intervals of the real numbers form a Helly family, if a family of closed intervals has an empty intersection and is minimal. A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X, every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. A disjoint union may mean one of two things, most simply, it may mean the union of sets that are disjoint
9.
Open (topology)
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In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of two subsets of a topological space are near without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces, in the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us a lot of information about points close to x =0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
10.
Subspace topology
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In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology. Given a topological space and a subset S of X, the topology on S is defined by τ S =. That is, a subset of S is open in the subspace topology if, if S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated, alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι, S ↪ X is continuous. More generally, suppose ι is an injection from a set S to a topological space X, then the subspace topology on S is defined as the coarsest topology for which ι is continuous. The open sets in topology are precisely the ones of the form ι −1 for U open in X. S is then homeomorphic to its image in X and ι is called a topological embedding. A subspace S is called a subspace if the injection ι is an open map. Likewise it is called a subspace if the injection ι is a closed map. The distinction between a set and a space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. In the following, R represents the numbers with their usual topology. The subspace topology of the numbers, as a subspace of R, is the discrete topology. The rational numbers Q considered as a subspace of R do not have the discrete topology. If a and b are rational, then the intervals and are open and closed. The set as a subspace of R is both open and closed, whereas as a subset of R it is only closed, as a subspace of R, ∪ is composed of two disjoint open subsets, and is therefore a disconnected space. Let S = [0, 1) be a subspace of the real line R, then [0, 1/2) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R, the subspace topology has the following characteristic property. Let Y be a subspace of X and let i, Y → X be the inclusion map, then for any topological space Z a map f, Z → Y is continuous if and only if the composite map i ∘ f is continuous. This property is characteristic in the sense that it can be used to define the topology on Y
11.
Disjoint sets
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In mathematics, two sets are said to be disjoint if they have no element in common. Equivalently, disjoint sets are sets whose intersection is the empty set, for example, and are disjoint sets, while and are not. This definition of disjoint sets can be extended to any family of sets, a family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint. For example, the collection of sets is pairwise disjoint, two sets are said to be almost disjoint sets if their intersection is small in some sense. For instance, two sets whose intersection is a finite set may be said to be almost disjoint. In topology, there are notions of separated sets with more strict conditions than disjointness. For instance, two sets may be considered to be separated when they have disjoint closures or disjoint neighborhoods, similarly, in a metric space, positively separated sets are sets separated by a nonzero distance. Disjointness of two sets, or of a family of sets, may be expressed in terms of their intersections, two sets A and B are disjoint if and only if their intersection A ∩ B is the empty set. It follows from definition that every set is disjoint from the empty set. A family F of sets is pairwise disjoint if, for two sets in the family, their intersection is empty. If the family more than one set, this implies that the intersection of the whole family is also empty. However, a family of one set is pairwise disjoint, regardless of whether that set is empty. Additionally, a family of sets may have an empty intersection without being pairwise disjoint, for instance, the three sets have an empty intersection but are not pairwise disjoint. In fact, there are no two disjoint sets in this collection, also the empty family of sets is pairwise disjoint. A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the intervals of the real numbers form a Helly family, if a family of closed intervals has an empty intersection and is minimal. A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X, every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. A disjoint union may mean one of two things, most simply, it may mean the union of sets that are disjoint
12.
Nonempty
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
13.
Open set
–
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of two subsets of a topological space are near without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces, in the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us a lot of information about points close to x =0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
14.
Empty set
–
In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
15.
Clopen set
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In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the meanings of open and closed are antonyms. A set is closed if its complement is open, which leaves the possibility of a set whose complement is also open. In any topological space X, the empty set and the whole space X are both clopen, now consider the space X which consists of the union of the two open intervals and of R. The topology on X is inherited as the topology from the ordinary topology on the real line R. In X, the set is clopen, as is the set and this is a quite typical example, whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen. As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, using the fact that 2 is not in Q, one can show quite easily that A is a clopen subset of Q. A topological space X is connected if and only if the only clopen sets are the empty set, a set is clopen if and only if its boundary is empty. Any clopen set is a union of connected components, if all connected components of X are open, then a set is clopen in X if and only if it is a union of connected components. A topological space X is discrete if and only if all of its subsets are clopen, using the union and intersection as operations, the clopen subsets of a given topological space X form a Boolean algebra. Every Boolean algebra can be obtained in this way from a topological space. Morris, Sidney A. Topology Without Tears, archived from the original on 19 April 2013
16.
Boundary (topology)
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In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, an element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set, notations used for boundary of a set S include bd, fr, and ∂S. Some authors use the term instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology. However, frontier sometimes refers to a different set, which is the set of points which are not actually in the set. A connected component of the boundary of S is called a component of S. If the set consists of points only, then the set has only a boundary. There are several definitions to the boundary of a subset S of a topological space X. The intersection of the closure of S with the closure of its complement, the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S. Consider the real line R with the usual topology, one has ∂ = ∂ = ∂ = ∂∅ = ∅ ∂Q = R ∂ = These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. In the space of rational numbers with the topology, the boundary of. The boundary of a set is a topological notion and may change if one changes the topology, for example, given the usual topology on R2, the boundary of a closed disk Ω = is the disks surrounding circle, ∂Ω =. If the disk is viewed as a set in R3 with its own usual topology, i. e. Ω =, then the boundary of the disk is the disk itself, ∂Ω = Ω. If the disk is viewed as its own space, then the boundary of the disk is empty. The boundary of a set is closed, the boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set. A set is the boundary of some open set if and only if it is closed, the boundary of a set is the boundary of the complement of the set, ∂S = ∂. The interior of the boundary of a set is the empty set. Hence, p is a point of a set if and only if every neighborhood of p contains at least one point in the set
17.
Maximal element
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In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are known, respectively. The maximum of a subset S of an ordered set is an element of S which is greater than or equal to any other element of S. While a partially ordered set can have at most one each maximum and minimum it may have multiple maximal and minimal elements, for totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an example, in the collection S = ordered by containment, the element is minimal, the element is maximal, the element is neither, Let be a partially ordered set and S ⊂ P. Then m ∈ S is an element of S if for all s ∈ S, m ≤ s implies m = s. The definition for minimal elements is obtained by using ≥ instead of ≤, example 1, Let S = [1, ∞) ⊂ ℝ, for all m∈S we have s=m+1∈S but m<s. Example 2, Let S = ⊂ ℚ and recall that 2 ∉ℚ, in general ≤ is only a partial order on S. If m is an element and s∈S, it remains the possibility that neither s≤m nor m≤s. This leaves open the possibility that there are many maximal elements, example 3, In the fence a1 < b1 > a2 < b2 > a3 < b3 >. all the ai are minimal, and all the bi are maximal, see picture. Example 4, Let A be a set with at least two elements and let S= be the subset of the power set P consisting of singletons, partially ordered by ⊂. This is the discrete poset—no two elements are comparable—and thus every element ∈S is maximal and for any distinct a‘, a‘‘ neither ⊂ nor ⊂. It looks like m should be a greatest element or maximum but in fact it is not necessarily the case, the definition of maximal element is somewhat weaker. Suppose we find s ∈ S with max S ≤ s, then, by the definition of greatest element, in other words, a maximum, if it exists, is the maximal element. The converse is not true, there can be maximal elements despite there being no maximum, example 3 is an instance of existence of many maximal elements and no maximum. The reason is, again, that in general ≤ is only an order on S. If m is an element and s ∈ S, it remains the possibility that neither s ≤ m nor m ≤ s
18.
Partition of a set
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In mathematics, a partition of a set is a grouping of the sets elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. A partition of a set X is a set of nonempty subsets of X such that every element x in X is in one of these subsets. Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold, the union of the sets in P is equal to X. The sets in P are said to cover X, the intersection of any two distinct sets in P is empty. The elements of P are said to be pairwise disjoint, the sets in P are called the blocks, parts or cells of the partition. The rank of P is |X| − |P|, if X is finite, every singleton set has exactly one partition, namely. The empty set ∅ has exactly one partition, namely ∅, for any nonempty set X, P = is a partition of X, called the trivial partition. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, the set has these five partitions, sometimes written 1|2|3. The following are not partitions of, is not a partition because one of its elements is the empty set, is not a partition because the element 2 is contained in more than one block. Is not a partition of because none of its blocks contains 3, however, thus the notions of equivalence relation and partition are essentially equivalent. The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly one element from each part of the partition and this implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class. Informally, this means that α is a fragmentation of ρ. In that case, it is written that α ≤ ρ and this finer-than relation on the set of partitions of X is a partial order. Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, the partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. These atomic partitions correspond one-for-one with the edges of a complete graph, in this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations, if D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes, the sets and. The 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes, and. In other words, given distinct numbers a, b, c in N, with a < b < c, if a ~ c, it follows that also a ~ b and b ~ c, that is b is also in C
19.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
20.
Clopen
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In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the meanings of open and closed are antonyms. A set is closed if its complement is open, which leaves the possibility of a set whose complement is also open. In any topological space X, the empty set and the whole space X are both clopen, now consider the space X which consists of the union of the two open intervals and of R. The topology on X is inherited as the topology from the ordinary topology on the real line R. In X, the set is clopen, as is the set and this is a quite typical example, whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen. As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, using the fact that 2 is not in Q, one can show quite easily that A is a clopen subset of Q. A topological space X is connected if and only if the only clopen sets are the empty set, a set is clopen if and only if its boundary is empty. Any clopen set is a union of connected components, if all connected components of X are open, then a set is clopen in X if and only if it is a union of connected components. A topological space X is discrete if and only if all of its subsets are clopen, using the union and intersection as operations, the clopen subsets of a given topological space X form a Boolean algebra. Every Boolean algebra can be obtained in this way from a topological space. Morris, Sidney A. Topology Without Tears, archived from the original on 19 April 2013
21.
Locally connected space
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In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. However, whereas the structure of compact subsets of Euclidean space was quite early on via the Heine–Borel theorem. Indeed, while any compact Hausdorff space is compact, a connected space –. As an example, the notion of weak local connectedness at a point, in the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifolds which are locally well understood but have complicated global behavior. By this it is meant that although the basic point-set topology of manifolds is relatively simple, local path connectedness will be discussed as well. A space is connected if and only if for every open set U. It follows, for instance, that a function from a locally connected space to a totally disconnected space must be locally constant. In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general, for instance Cantor space is totally disconnected, Let X be a topological space, and let x be a point of X. We say that X is locally connected at x if for every open set V containing x there exists a connected, the space X is said to be locally connected if it is locally connected at x for all x in X. Note that local connectedness and connectedness are not related to one another, by contrast, we say that X is weakly locally connected at x if for every open set V containing x there exists a connected subset N of V such that x lies in the interior of N. An equivalent definition is, each open set V containing x contains an open neighborhood U of x such that any two points in U lie in some connected subset of V. The space X is said to be locally connected if it is weakly locally connected at x for all x in X. Evidently a space which is connected at x is weakly locally connected at x. We say that X is locally connected at x if for every open set V containing x there exists a path connected. The space X is said to be path connected if it is locally path connected at x for all x in X. Since path connected spaces are connected, locally connected spaces are locally connected. This time the converse does not hold, for any positive integer n, the Euclidean space R n is locally path connected, thus locally connected, it is also connected
22.
Open sets
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In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of two subsets of a topological space are near without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces, in the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us a lot of information about points close to x =0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
23.
Hausdorff space
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In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space and it implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology, hausdorffs original definition of a topological space included the Hausdorff condition as an axiom. Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of x, X is a Hausdorff space if all distinct points in X are pairwise neighborhood-separable. This condition is the separation axiom, which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used, a related, but weaker, notion is that of a preregular space. X is a space if any two topologically distinguishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces, the relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular and Kolmogorov, a topological space is preregular if and only if its Kolmogorov quotient is Hausdorff. For a topological space X, the following are equivalent, X is a Hausdorff space, limits of nets in X are unique. Limits of filters on X are unique, any singleton set ⊂ X is equal to the intersection of all closed neighbourhoods of x. The diagonal Δ = is closed as a subset of the product space X × X, almost all spaces encountered in analysis are Hausdorff, most importantly, the real numbers are a Hausdorff space. More generally, all spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups, a simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set. Pseudometric spaces typically are not Hausdorff, but they are preregular, indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff. The related concept of Scott domain also consists of non-preregular spaces, while the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit. Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff, in fact, every topological space can be realized as the quotient of some Hausdorff space. Hausdorff spaces are T1, meaning that all singletons are closed, another nice property of Hausdorff spaces is that compact sets are always closed
24.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
25.
Convex set
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In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations. For example, a cube is a convex set, but anything that is hollow or has an indent, for example. The boundary of a set is always a convex curve. The intersection of all convex sets containing a given subset A of Euclidean space is called the hull of A. It is the smallest convex set containing A, a convex function is a real-valued function defined on an interval with the property that its epigraph is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets, the branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. The notion of a set can be generalized as described below. Let S be a space over the real numbers, or, more generally. A set C in S is said to be if, for all x and y in C and all t in the interval. In other words, every point on the segment connecting x and y is in C. This implies that a set in a real or complex topological vector space is path-connected. Furthermore, C is strictly convex if every point on the segment connecting x and y other than the endpoints is inside the interior of C. A set C is called convex if it is convex. The convex subsets of R are simply the intervals of R, some examples of convex subsets of the Euclidean plane are solid regular polygons, solid triangles, and intersections of solid triangles. Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids, the Kepler-Poinsot polyhedra are examples of non-convex sets. A set that is not convex is called a non-convex set, the complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. If S is a set in n-dimensional space, then for any collection of r, r >1. Ur in S, and for any nonnegative numbers λ1, + λr =1, then one has, ∑ k =1 r λ k u k ∈ S
26.
Simply connected set
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If a space is not simply-connected, it is convenient to measure the extent to which it fails to be simply-connected, this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space, if there are no holes, the group is trivial — equivalently. Informally, an object in our space is simply-connected if it consists of one piece. For example, neither a doughnut nor a cup is simply connected. In two dimensions, a circle is not simply-connected, but a disk and a line are, spaces that are connected but not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected. To illustrate the notion of connectedness, suppose we are considering an object in three dimensions, for example, an object in the shape of a box, a doughnut. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a piece of string and trails it through the water inside the aquarium, in whatever way he pleases. Now the loop begins to contract on itself, getting smaller and smaller, if the loop can always shrink all the way to a point, then the aquariums interior is simply connected. If sometimes the loop gets caught — for example, around the hole in the doughnut — then the object is not simply-connected. Notice that the only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point. The stronger condition, that the object has no holes of any dimension, is called contractibility, intuitively, this means that p can be continuously deformed to get q while keeping the endpoints fixed. Hence the term simply connected, for any two points in X, there is one and essentially only one path connecting them. A third way to express the same, X is simply-connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i. e. consists only of the identity element. Yet another formulation is used in complex analysis, an open subset X of C is simply-connected if. It might also be worth pointing out that a relaxation of the requirement that X be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a set has connected extended complement exactly when each of its connected components are simply-connected
27.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
28.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines
29.
Cantor set
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In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883, through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. Although Cantor himself defined the set in a general, abstract way, Cantor himself mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. The Cantor ternary set C is created by deleting the open middle third from a set of line segments. One starts by deleting the open middle third from the interval, next, the open middle third of each of these remaining segments is deleted, leaving four line segments, ∪ ∪ ∪. This process is continued ad infinitum, where the nth set is C n = C n −13 ∪ for n ≥1, and C0 =. The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process, the first six steps of this process are illustrated below. This process of removing middle thirds is an example of a finite subdivision rule. It is perhaps most intuitive to think about the Cantor set as the set of numbers between zero and one whose ternary expansion in base three doesnt contain the digit 1. As the above shows, the Cantor ternary set is in bijection with the set of paths in a full binary tree on countably many nodes. Such a path is determined by an infinite series of instructions determining at each node whether we go left or right as we traverse the diagram. This in turn describes the expansion of the number. For example, such a path might begin which describes the ternary number 0.02200, in particular, the Cantor set is canonically in bijection with the set of binary sequences. Since the Cantor set is defined as the set of points not excluded and this total is the geometric progression ∑ n =0 ∞2 n 3 n +1 =13 +29 +427 +881 + ⋯ =13 =1. So that the left is 1 –1 =0. This calculation shows that the Cantor set cannot contain any interval of non-zero length, in fact, it may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a look at the process reveals that there must be something left. So removing the segment from the original interval leaves behind the points 1/3 and 2/3
30.
Homotopy
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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. If we think of the parameter of H as time then H describes a continuous deformation of f into g, at time 0 we have the function f. We can also think of the second parameter as a control that allows us to smoothly transition from f to g as the slider moves from 0 to 1. The two versions coincide by setting ht = H and it is not sufficient to require each map ht to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. The animation shows the image of ht as a function of the parameter t and it pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. Continuous functions f and g are said to be homotopic if, being homotopic is an equivalence relation on the set of all continuous functions from X to Y. The maps f and g are called homotopy equivalences in this case, every homeomorphism is a homotopy equivalence, but the converse is not true, for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent. As another example, the Möbius strip and an untwisted strip are homotopy equivalent, spaces that are homotopy equivalent to a point are called contractible. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations, the first example of a homotopy equivalence is R n with a point, denoted R n ≃. There is an equivalence between S1 and R2 −. More generally, R n − ≃ S n −1, any fiber bundle π, E → B with fibers F b homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since π, R n − → S n −1 is a bundle with fiber R >0. Every vector bundle is a bundle with a fiber homotopy equivalent to a point. For any 0 ≤ k < n, R n − R k ≃ S n − k −1 by writing R n as R k × R n − k, a function f is said to be null-homotopic if it is homotopic to a constant function. For example, a map f from the unit circle S1 to any space X is null-homotopic precisely when it can be extended to a map from the unit disk D2 to X that agrees with f on the boundary
31.
Topologist's sine curve
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In the branch of mathematics known as topology, the topologists sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the sin on the half-open interval (0, 1], together with the origin, under the topology induced from the Euclidean plane. As x approaches zero from the right, the magnitude of the rate of change of 1/x increases and this is why the frequency of the sine wave increases as one moves to the left in the graph. The topologists sine curve T is connected but neither connected nor path connected. This is because it includes the point but there is no way to link the function to the origin so as to make a path. The space T is the image of a locally compact space (namely, let V be the space ∪. The topological dimension of T is 1, two variants of the topologists sine curve have other interesting properties. The closed topologists sine curve can be defined by taking the topologists sine curve, the extended topologists sine curve can be defined by taking the closed topologists sine curve and adding to it the set. It is arc connected but not locally connected, steen, Lynn Arthur, Seebach, J. Arthur Jr. Counterexamples in Topology, Mineola, NY, Dover Publications, Inc. pp. 137–138, ISBN 978-0-486-68735-3, MR1382863 Weisstein, Eric W. Topologists Sine Curve
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General linear group
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two matrices is again invertible, and the inverse of an invertible matrix is invertible. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix, for example, the general linear group over R is the group of n×n invertible matrices of real numbers, and is denoted by GLn or GL. More generally, the linear group of degree n over any field F, or a ring R, is the set of n×n invertible matrices with entries from F. Typical notation is GLn or GL, or simply GL if the field is understood, more generally still, the general linear group of a vector space GL is the abstract automorphism group, not necessarily written as matrices. The special linear group, written SL or SLn, is the subgroup of GL consisting of matrices with a determinant of 1, the group GL and its subgroups are often called linear groups or matrix groups. These groups are important in the theory of representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general. The modular group may be realised as a quotient of the linear group SL. If n ≥2, then the group GL is not abelian, if V has finite dimension n, then GL and GL are isomorphic. The isomorphism is not canonical, it depends on a choice of basis in V, in a similar way, for a commutative ring R the group GL may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL for any R-module, but in general this is not isomorphic to GL, over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore, a definition of GL is as the group of matrices with nonzero determinant. Over a non-commutative ring R, determinants are not at all well behaved, in this case, GL may be defined as the unit group of the matrix ring M. The general linear group GL over the field of numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn, the subset GL consists of those matrices whose determinant is non-zero. The determinant is a map, and hence GL is an open affine subvariety of Mn. The Lie algebra of GL, denoted g l n, consists of all n×n real matrices with the serving as the Lie bracket. As a manifold, GL is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant, the identity component, denoted by GL+, consists of the real n×n matrices with positive determinant
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Finitely generated projective module
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Various equivalent characterizations of these modules appear below. A free module is a module, but the converse may be wrong over some rings. However, every module is a free module over a principal ideal domain. Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan, the usual definition in line with category theory is the property of lifting that carries over from free to projective modules. The advantage of this definition of projective is that it can be carried out in more general than module categories. It can also be dualized, leading to injective modules, a module P is projective if and only if for every surjective module homomorphism f, M ↠ P there exists a module homomorphism h, P → M such that fh = idP. The existence of such a map h implies that P is a direct summand of M. More explicitly, M = im ⊕ ker, and im is isomorphic to P. The foregoing is a description of the following statement, A module P is projective if every short exact sequence of modules of the form 0 → A → B → P →0 is a split exact sequence. A module P is projective if and only if there is a free module F and another module Q such that the direct sum of P and Q is F. An R-module P is projective if and only if the functor Hom, R-Mod → AB is an exact functor, when the ring R is commutative, AB is advantageously replaced by R-Mod in the preceding characterization. This functor is left exact, but, when P is projective. This means that P is projective if and only if this functor preserves epimorphisms, or if it preserves finite colimits. A module P is projective if and only if there exists a set and a set such that for x in P, fi is only nonzero for finitely many i. The following properties of projective modules are quickly deduced from any of the definitions of projective modules, Direct sums. If e = e2 is an idempotent in the ring R, the right-to-left implications are true over the rings labeling them. There may be other rings over which they are true, for example the implication labeled local ring or PID is also true for polynomial rings over a field, this is Quillen–Suslin theorem. The converse is true in the cases, if R is a field or skew field
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Annulus (mathematics)
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In mathematics, an annulus is a ring-shaped object, a region bounded by two concentric circles. The open annulus is topologically equivalent to both the open cylinder S1 × and the punctured plane, informally, it has the shape of a hardware washer. The area of an annulus is the difference in the areas of the circle of radius R. The area of an annulus is determined by the length of the longest line segment within the annulus, the area of an annulus sector of angle θ, with θ measured in radians, is given by A = θ2. In complex analysis an annulus ann in the plane is an open region defined as r < | z − a | < R. If r is 0, the region is known as the disk of radius R around the point a. As a subset of the plane, an annulus can be considered as a Riemann surface. The complex structure of an annulus depends only on the ratio r/R, each annulus ann can be holomorphically mapped to a standard one centered at the origin and with outer radius 1 by the map z ↦ z − a R. The inner radius is then r/R <1, the Hadamard three-circle theorem is a statement about the maximum value a holomorphic function may take inside an annulus
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Disk (mathematics)
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In geometry, a disk is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not. In Cartesian coordinates, the disk of center and radius R is given by the formula D = while the closed disk of the same center. The area of a closed or open disk of radius R is πR2, the open disk and the closed disk are not topologically equivalent, as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact, however from the viewpoint of algebraic topology they share many properties, both of them are contractible and so are homotopy equivalent to a single point. This implies that their groups are trivial, and all homology groups are trivial except the 0th one. The Euler characteristic of a point is 1, every continuous map from the closed disk to itself has at least one fixed point, this is the case n=2 of the Brouwer fixed point theorem. The statement is false for the disk, Consider for example the function f = which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the unit disk fixes every point on the half circle x 2 + y 2 =1, x >0
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Subspace (topology)
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In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology. Given a topological space and a subset S of X, the topology on S is defined by τ S =. That is, a subset of S is open in the subspace topology if, if S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated, alternatively we can define the subspace topology for a subset S of X as the coarsest topology for which the inclusion map ι, S ↪ X is continuous. More generally, suppose ι is an injection from a set S to a topological space X, then the subspace topology on S is defined as the coarsest topology for which ι is continuous. The open sets in topology are precisely the ones of the form ι −1 for U open in X. S is then homeomorphic to its image in X and ι is called a topological embedding. A subspace S is called a subspace if the injection ι is an open map. Likewise it is called a subspace if the injection ι is a closed map. The distinction between a set and a space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. In the following, R represents the numbers with their usual topology. The subspace topology of the numbers, as a subspace of R, is the discrete topology. The rational numbers Q considered as a subspace of R do not have the discrete topology. If a and b are rational, then the intervals and are open and closed. The set as a subspace of R is both open and closed, whereas as a subset of R it is only closed, as a subspace of R, ∪ is composed of two disjoint open subsets, and is therefore a disconnected space. Let S = [0, 1) be a subspace of the real line R, then [0, 1/2) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R, the subspace topology has the following characteristic property. Let Y be a subspace of X and let i, Y → X be the inclusion map, then for any topological space Z a map f, Z → Y is continuous if and only if the composite map i ∘ f is continuous. This property is characteristic in the sense that it can be used to define the topology on Y
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Path (topology)
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In mathematics, a path in a topological space X is a continuous function f from the unit interval I = to X f, I → X. The initial point of the path is f and the point is f. One often speaks of a path from x to y where x and y are the initial and terminal points of the path, note that a path is not just a subset of X which looks like a curve, it also includes a parameterization. For example, the maps f = x and g = x2 represent two different paths from 0 to 1 on the real line, a loop in a space X based at x ∈ X is a path from x to x. A loop may be well regarded as a map f, I → X with f = f or as a continuous map from the unit circle S1 to X f. This is because S1 may be regarded as a quotient of I under the identification 0 ∼1, the set of all loops in X forms a space called the loop space of X. A topological space for which exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components, the set of path-connected components of a space X is often denoted π0. One can also define paths and loops in pointed spaces, which are important in homotopy theory, if X is a topological space with basepoint x0, then a path in X is one whose initial point is x0. Likewise, a loop in X is one that is based at x0, paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed. Specifically, a homotopy of paths, or path-homotopy, in X is a family of paths ft, I → X indexed by I such that ft = x0, the map F, I × I → X given by F = ft is continuous. The paths f0 and f1 connected by a homotopy are said to homotopic, one can likewise define a homotopy of loops keeping the base point fixed. The relation of being homotopic is a relation on paths in a topological space. The equivalence class of a path f under this relation is called the class of f. One can compose paths in a space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z. The path fg is defined as the path obtained by first traversing f, clearly path composition is only defined when the terminal point of f coincides with the initial point of g