Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
Zorns Lemma (film)
Zorns Lemma is a 1970 American structural experimental film by Hollis Frampton. Starting as a series of photographs, the non-narrative film is structured around a 24-letter Latin alphabet, it remains, along with Michael Snow's Wavelength and Tony Conrad's The Flicker, one of the best known examples of structural filmmaking. The opening section of Zorns Lemma is 5 minutes long. In it a woman reads an abecedary of 24 couplets from The Bay State Primer, an eighteenth century book designed to teach children the alphabet; the film is black during this section. The film's main section lasts 45 minutes, broken into 2,700 one-second units, it shows the viewer a 24-part "alphabet" that evolves. The section begins by presenting each letter typed on a sheet of tin foil; the alphabet is composed of words that appear on street signs, photographed in Manhattan. As the film continues to cycle through the alphabet, individually filmed letters are substituted with abstract moving images; the first four substitutions—fire, waves and reeds –depict the four classical elements, by the end, it is composed of moving images which represent them.
The film's conclusion lasts 10 minutes. It shows a man and dog walking through snow. Six women's voices alternate in reading the words of a passage from Robert Grosseteste's medieval document On Light, or the Ingression of Forms, which Frampton translated and edited for the film; the voices read the text at a rate of one word per second. Zorns Lemma emerged from Word Pictures, a photography project that Frampton made from 1962 to 1963. For Word Pictures, Frampton shot over 2000 black-and-white 35mm photographs of environmental words, seeking to explore the illusions of photography as a medium. However, he had difficulty devising a form in. During this period, Frampton became less active as a photographer and first started to experiment with filmmaking, he started to make a film by shooting the photographs on a stand but thought the result looked "dead", deciding that he should film the words in color. He began filming static shots of words framing them with the use of a tripod. Frampton thought that imposing his own order on the film would make it too similar to a poem, so he decided to use alphabetical order and avoid any conscious connections between words.
Because the number of times he had captured each letter varied he decided to use the "double alphabet" and continuously cycle through the letters. This structure left "holes" to be filled in the film. Frampton decided to substitute out the letters for images planning to use different images each time. Frampton's production notes outline three criteria for selecting the images used, he preferred banal images, with the exception of animals included for "shock value". He preferred "sculptural" images showing work or the "illusion of space or substance"; these included changing a tire. Frampton worked in "cinematic or para-cinematic reference" subtly so; the letter d is substituted with cookie dough being sliced with a star-shaped cookie cutter, as a cinematic reference to film "stars". Modular actions—such as a child swinging or assembling Tinkertoys —serve as para-cinematic references to modular time. Frampton set 24 frames. However, 24 of the shots have slight irregularities in their lengths. Frampton instructed key grip David Hamilton that 12 shots should have 23 frames and 12 shots should have 25 frames.
The final section of Zorns Lemma was shot in 1970. Frampton envisioned it as a time for the audience "to empty the mind"; when loading and unloading the film rolls, he intentionally exposed them to sunlight and created reel-end flares to emphasize "the materiality of film". Frampton's films are titled after specialized scientific disciplines, as in the case of Maxwell's Demon, Prince Rupert's Drops, Hapax Legomena. In the early 1960s minimalist artist Carl Andre described to Frampton the Dedekind cut, which partitions a ordered set into a set with no maximal element and a set with no minimal element, he became interested in the relationship between set theory and film while working on his ongoing project Magellan. Frampton titled Zorns Lemma after Zorn's lemma, a proposition of set theory formulated by mathematician Max Zorn in 1935. Zorn's lemma describes ordered sets where every ordered subset has an upper bound; the letters and images in Zorns Lemma are sets whose order is discovered during the course of the film.
Zorns Lemma premiered at the Philharmonic Hall for the 1970 New York Film Festival. It was the first experimental feature film. J. Hoberman wrote that it "drove the audience mad", Howard Thompson observed that "never, at least so far during the Film Festival, have so many Philharmonic Hall viewers slithered outside for a cigarette." The sale of Zorns Lemma was a financial success for Frampton. Amos Vogel called Zorns Lemma a "radical example of reductive cinema" that warned of "things to come...'meaning' has been eliminated and the work exists purely for itself, demanding attention to structure and orchestration." Kevin Thomas of the Los Angeles Times described the film as "a demanding but stimulating exercise in heightening our awareness of the possibilities of visual perception and, the ways in which we create meaning itself". The experimental filmmaker Ernie Gehr stated, "Zorns Lemma is a major poetic work. Created and put together by a clear eye, this original and complex abstract work moves beyond the letters of the alphabet, beyon
John Wilder Tukey was an American mathematician best known for development of the FFT algorithm and box plot. The Tukey range test, the Tukey lambda distribution, the Tukey test of additivity, the Teichmüller–Tukey lemma all bear his name, he is credited with coining the term'bit'. Tukey was born in New Bedford, Massachusetts, in 1915, obtained a B. A. in 1936 and M. Sc. in 1937, in chemistry, from Brown University, before moving to Princeton University where he received a Ph. D. in mathematics. During World War II, Tukey worked at the Fire Control Research Office and collaborated with Samuel Wilks and William Cochran. After the war, he returned to Princeton, dividing his time between the university and AT&T Bell Laboratories, he became a full professor at 35 and founding chairman of the Princeton statistics department in 1965. Among many contributions to civil society, Tukey served on a committee of the American Statistical Association that produced a report challenging the conclusions of the Kinsey Report, Statistical Problems of the Kinsey Report on Sexual Behavior in the Human Male.
He was awarded the National Medal of Science by President Nixon in 1973. He was awarded the IEEE Medal of Honor in 1982 "For his contributions to the spectral analysis of random processes and the fast Fourier transform algorithm." Tukey retired in 1985. He died in New Brunswick, New Jersey, on July 26, 2000. Early in his career Tukey worked on developing statistical methods for computers at Bell Labs where he invented the term "bit", his statistical interests were many and varied. He is remembered for his development with James Cooley of the Cooley–Tukey FFT algorithm. In 1970, he contributed to what is today known as the jackknife estimation—also termed Quenouille–Tukey jackknife, he introduced the box plot in his 1977 book, "Exploratory Data Analysis." Tukey's range test, the Tukey lambda distribution, Tukey's test of additivity, Tukey's lemma, the Tukey window all bear his name. He is the creator of several little-known methods such as the trimean and median-median line, an easier alternative to linear regression.
In 1974, he developed, with the concept of the projection pursuit. He contributed to statistical practice and articulated the important distinction between exploratory data analysis and confirmatory data analysis, believing that much statistical methodology placed too great an emphasis on the latter. Though he believed in the utility of separating the two types of analysis, he pointed out that sometimes in natural science, this was problematic and termed such situations uncomfortable science. A. D. Gordon offered the following summary of Tukey's principles for statistical practice:... the usefulness and limitation of mathematical statistics. Tukey coined many statistical terms that have become part of common usage, but the two most famous coinages attributed to him were related to computer science. While working with John von Neumann on early computer designs, Tukey introduced the word "bit" as a contraction of "binary digit"; the term "bit" was first used in an article by Claude Shannon in 1948.
In 2000, Fred Shapiro, a librarian at the Yale Law School, published a letter revealing that Tukey's 1958 paper "The Teaching of Concrete Mathematics" contained the earliest known usage of the term "software" found in a search of JSTOR's electronic archives, predating the OED's citation by two years. This led many to credit Tukey with coining the term in obituaries published that same year, although Tukey never claimed credit for any such coinage. In 1995, Paul Niquette claimed he had coined the term in October 1953, although he could not find any documents supporting his claim; the earliest known publication of the term "software" in an engineering context was in August 1953 by Richard R. Carhart, in a Rand Corporation Research Memorandum. List of pioneers in computer science Andrews, David F. Robust estimates of location: survey and advances. Princeton University Press. ISBN 978-0-691-08113-7. OCLC 369963. Basford, Kaye E. Graphical analysis of multiresponse data. Chapman & Hall/CRC. ISBN 978-0-8493-0384-5.
OCLC 154674707. Blackman, R B; the measurement of power spectra from the point of view of communications engineering. Dover Publications. ISBN 978-0-486-60507-4. Cochran, William G. Statistical problems of the Kinsey report on sexual behavior in the human male. Journal of the American Statistical Association. Hoaglin, David C. Understanding Robust and Exploratory Data Analysis. Wiley. ISBN 978-0-471-09777-8. OCLC 8495063. CS1 maint: Multiple names: authors list CS1 maint: Extra text: authors list Hoaglin, David C. Exploring Data Tables and Shapes. Wiley. ISBN 978-0-471-09776-1. OCLC 11550398. CS1 maint: Multiple names: authors list CS1 maint: Extra text: authors list Hoagl
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Kazimierz Kuratowski was born in Warsaw, Vistula Land, on 2 February 1896, into an assimilated Jewish family, he was a son of Marek Kuratow, a barrister, Róża Karzewska. He completed a Warsaw secondary school, named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian, he completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics, he obtained his Ph. D. in 1921, in newly independent Poland. In autumn 1921 Kuratowski was awarded the Ph. D. degree for his groundbreaking work. His thesis statement consisted of two parts.
One was devoted to an axiomatic construction of topology via the closure axioms. This first part has been cited in hundreds of scientific articles; the second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Stefan Mazurkiewicz. Kuratowski's thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin. Two years in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University, he was appointed a full professor of mathematics at Lwów Polytechnic in Lwów, in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific Society While Kuratowski associated with many of the scholars of the Lwów School of Mathematics, such as Stefan Banach and Stanislaw Ulam, the circle of mathematicians based around the Scottish Café he kept close connections with Warsaw.
Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun, hence did not contribute any problems to it. He did however, collaborate with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University. A year Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. During World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968.
After World War II, Kuratowski was involved in the rebuilding of scientific life in Poland. He helped to establish the State Mathematical Institute, incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, was a long-time chairman of the Polish and International Mathematics Societies, he was president of the Scientific Council of the State Institute of Mathematics. From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski. Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów's Scottish Café, he was a member of the Warsaw Scientific Society. What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin, he was one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences.
High quality research monographs of the representatives of Warsaw's and Lwów’s School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, including the Royal Society of Edinburgh, Germany, Hungary and the Union of Soviet Socialist Republics. In 1981, IMPAN, the Polish Mathematical Society, Kuratowski's daughter Zofia Kuratowska established a prize in his name for achievements in mathematics to people under the age of 30 years; the prize is considered the most prestigious of awards for young Polish mathematicians. Kuratowski’s research focused on abstract topological and metric structures, he implemented the closure axioms. This was fundamental for the development of topological space theory and irreducible continuum theory between two points; the most valuable results, which were obtained by Kazimierz K
Nicolas Bourbaki is the collective pseudonym of a group of mathematicians. Their aim is to reformulate mathematics on an abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality, their work led to the discovery of several concepts and terminologies still used, influenced modern branches of mathematics. While there is no one person named Nicolas Bourbaki, the Bourbaki group known as the Association des collaborateurs de Nicolas Bourbaki, has an office at the École Normale Supérieure in Paris. In 1934, young French mathematicians from various French universities felt the need to form a group to jointly produce textbooks that they could all use for teaching. André Weil organized the first meeting on 10 December 1934 in the basement of a Parisian grill room, while all participants were attending a conference in Paris. Accounts of the early days vary; the founding members were all connected to the École Normale Supérieure in Paris and included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, René de Possel, Szolem Mandelbrojt and André Weil.
There was a preliminary meeting, towards the end of 1934. Jean Leray and Paul Dubreil were present at the preliminary meeting but dropped out before the group formed. Other notable participants in days were Hyman Bass, Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, Jean-Louis Koszul, Samuel Eilenberg, Serge Lang and Roger Godement; the original goal of the group had been to compile an improved mathematical analysis text. There was no official status of membership, at the time the group was quite secretive and fond of supplying disinformation. Regular meetings were scheduled, during which the group would discuss vigorously every proposed line of every book. Members had to resign by age 50, which resulted in a complete change of personnel by 1958. However, historian Liliane Beaulieu was quoted as never having found written affirmation of this rule; the atmosphere in the group can be illustrated by an anecdote told by Laurent Schwartz. Dieudonné and spectacularly threatened to resign unless topics were treated in their logical order, after a while others played on this for a joke.
Godement's wife wanted to see Dieudonné announcing his resignation, so on one occasion while she was there Schwartz deliberately brought up again the question of permuting the order in which measure theory and topological vector spaces were to be handled, to precipitate a guaranteed crisis. The name "Bourbaki" refers to Charles Denis Bourbaki, it is said. This is less confirmed by Robert Mainard; the Bourbaki group released. For example, the group released a wedding announcement, relating the marriage of Betti Bourbaki with a certain Hector Pétard. In November 1968, a mock obituary of Nicolas Bourbaki was released during one of the seminars, containing a few mathematical puns; the group is however still active as of 2018, organizing seminars and having released a book in 2016. Bourbaki's main work is the Elements of Mathematics series; this series aims to be a self-contained treatment of the core areas of modern mathematics. Assuming no special knowledge of mathematics, it takes up mathematics from the beginning, proceeds axiomatically and gives complete proofs.
The dates indicated below are for the first edition of the first chapter of each book. Most of the books were reedited several times, the books were released in several parts containing different chapters. Bourbaki, Nicolas. Livre I: Théorie des ensembles. Bourbaki, Nicolas. Livre II: Algèbre. Bourbaki, Nicolas. Livre III: Topologie. Bourbaki, Nicolas. Livre IV: Fonctions d'une variable réelle. Bourbaki, Nicolas. Livre V: Espaces vectoriels topologiques. Bourbaki, Nicolas. Livre VI: Intégration. Bourbaki, Nicolas. Livre VII: Algèbre commutative. Bourbaki, Nicolas. Livre VIII: Groupes et algèbres de Lie. Bourbaki, Nicolas. Livre IX: Théories spectrales. Bourbaki, Nicolas. Livre X: Variétés différentielles et analytiques. Bourbaki, Nicolas. Livre XI: Topologie algébrique; the book Variétés différentielles et analytiques was a fascicule de résultats, that is, a summary of results, on the theory of manifolds, rather than a worked-out exposition. The volume on spectral theory from 1967 was for four decades the last new book to be added to the series.
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. 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. 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; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. 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, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and 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 special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t