In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity and convergence. 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 and appear in every branch of modern mathematics; the branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Around 1735, Euler discovered the formula V − E + F = 2 relating the number of vertices and faces of a convex polyhedron, hence of a planar graph; the study and generalization of this formula by Cauchy and L'Huilier, is at the origin of topology.
In 1827, Carl Friedrich Gauss published General investigations of curved surfaces which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitely small distance from A are deflected infinitely little from one and the same plane passing through A."Yet, "until Riemann’s work in the early 1850s, surfaces were always dealt with from a local point of view and topological issues were never considered." "Möbius and Jordan seem to be the first to realize that the main problem about the topology of surfaces is to find invariants to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not."The subject is defined by Felix Klein in his "Erlangen Program": the geometry invariants of arbitrary continuous transformation, a kind of geometry.
The term "topology" was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Poincaré, his first article on this topic appeared in 1894. In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space, locally like a Euclidean plane; the utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application; the most used is that in terms of open sets, but more intuitive is that in terms of neighbourhoods and so this is given first. This axiomatization is due to Felix Hausdorff. Let X be a set. We allow X to be empty. 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 N is a neighbourhood of x 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 N is 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 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, where a subset N of R is defined to be a neighbourhood of a real number x if it includes an open interval containing x. Given such a structure, a subset U of X is defined to be open if U is a neighbourhood of all points in U.
The open sets satisfy the axioms given below. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining N to be a neighbourhood of x if N includes an open set U such that x ∈ U. A topological space is an ordered pair, where X is a set and τ is a collection of subsets of X, satisfying the following axioms: The empty set and X itself belong to τ. Any arbitrary 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. Given X = Z, the set of integers, the collection τ of all finite subsets of the integers plus Z itself is not a topology, because the union of all finite sets not containing zero is infinite but is not all of Z, so is not in τ.
Using de Morgan's laws
Israel Cristostomo "Izzy" Alcántara is a former Major League Baseball player for the Boston Red Sox and Milwaukee Brewers. Alcántara was signed as an amateur free agent by the Montreal Expos in 1990. In the minor leagues, Alcantara was successful, with at least 27 home runs and an OPS of.940 or higher in each of his seasons from 1997 to 2001. He led the International League in home runs twice, with 36 in 2001 and 27 in 2002, his OPS of 1.023 in 2000 was the best in the International League. In all his minor league years he batted.283 with 239 home runs and 988 runs batted in. Alcántara's minor league success did not carry over to the little time he spent in the Major Leagues, as he played parts of two seasons for the Boston Red Sox, debuting in the major leagues on June 25, 2000; that season, he hit.289 with four home runs in 45 at bats. Despite some success at the plate, Alcántara found himself deep on the Red Sox bench after irking Boston manager Jimy Williams by a perceived lack of hustle during a game against the Chicago White Sox.
For a time, Williams refused to play Alcántara despite general manager Dan Duquette's insistence the outfielder be put in the lineup of a team that suffered from low power numbers. In 2001, Alcántara's average declined to.263 with no home runs in 38 at bats in Boston although he continued to smash AAA pitching. Alcántara is remembered for an incident while he was playing with the Pawtucket Red Sox, Boston's AAA affiliate of the International League. On July 3, 2001, after being brushed back a second time by Scranton/Wilkes-Barre Red Barons pitcher Blas Cedeño, Alcántara turned around, kicking catcher Jeremy Salazar, karate-style, in the mask before charging the mound, throwing an unsuccessful punch at Cedeño before turning around as other players joined the fight until he was tackled by Kevin Orie; the incident cost Alcántara a six-game suspension and his spot on the International League's All Star squad. Before the 2002 season, he was signed as a free agent by the Milwaukee Brewers, where he hit.250 with 2 home runs in 32 at bats.
His final Major League game was on August 22, 2002 and he was released by the team after that season. Alcántara played in 2003 with the Mexican League team Vaqueros Laguna but after agreeing a release with his Mexican team, he moved to the Korean league after he played 61 games with 14 home runs and a.341 batting average. He signed a US$100,000 contract with LG Twins including a US$30,000 signing bonus, he came back to Vaqueros Laguna for the 2004 season, but this time Vaqueros arranged a high buyout clause to his contract in order to have him the whole season. On 24 March 2004 he tied a Mexican league record after hitting 8 home runs in six games, tying the previous record set by the Mexican Héctor Espino in 1969, he was selected to play the Mexican home run derby where he made it to the final round losing to the American Morgan Burkhart 7-8. He finished the season leading the Mexican league in home runs with 27. After the Mexican league imposed a salary cap starting in the 2004 season, that Alcántara refused to accept, his was suspended.
He returned to the Korean baseball for the 2004 season with Doosan Bears. Playing the Korean playoff against the Kia Tigers and winning 2-0 but losing to Samsung Lions 1–3 and winning the league third place, he played for Uni-President Lions of Taiwan's Chinese Professional Baseball League in 2005, where he was nicknamed "Al-Qaeda". He hit 15 home-runs in 44 games, with a batting average of.330, but was released for reasons never explained by the management. Playing in Isolation: A History of Baseball in Taiwan, p. 153, at Google Books He played with the Mexican team Rojos del Águila de Veracruz, posting a.281 with five home runs and 15 RBI in just 27 games. He joined Rojos del Águila de Veracruz in July in the late part for the league playoffs fight but his team was disqualified by Tigres de la Angelópolis. Alcántara played four games for the New Jersey Jackals of the independent Canadian-American Association of Professional Baseball, homering on 10 August before being released on 15 August.
Alcántara contributed with the Colombian team Caimanes de Barranquilla 2007–08 league championship, when he hit a home run in the 10th inning of the fourth game of the best-of-7 series that his team swept. Alcantara played for the Dominican team Tigres del Licey from 1994–95 to the 2004–05 season, winning the 2004 Caribbean Series and the All-Star team selection, he led the Dominican Professional Baseball League in home runs in the 1999-00 season with six, had 10 HR with 34 RBI and the 2002–03 when he hit 9 HR and had produced 21 RBI. He was released by Licey after hitting.171 with only 4 homers, but surpassing Alonzo Perry with 47 home runs with the Licey uniform and played with Leones del Escogido in the 2005–06 and with Estrellas Orientales and for Gigantes del Cibao for the 2006-07 season, retiring in fourth place with 52 HR in the Dominican league. He acted as assistant coach for Tigres del Licey in the 2009–10 season. Since 2015, Alcantara become a Christian. Career statistics and player information from Baseball-Reference, or Fangraphs, or Baseball-Reference
Winnefeld is an abandoned village in the district of Northeim in Lower Saxony, Germany. The site of the former village lies inmidst the Solling mountain range right at the Bundesstraße 241, where the ground ist made up of Loess; the nearest populated places are Nienover. According to results of archaeological digs by Hans-Georg Stephan and soil studies, the site used to be a wood pasture in the 8th century. Between 1150 and 1250, the Counts of Dassel had the woods cut area-wide, thus allowing agriculture and settlement of people; some 25 to 40 farms developed there to provide Nienover with food. The village was destroyed in 1342 by water of the Reiherbach amidst the St. Mary Magdalene's flood; the village had a church, built of stone in the years around 1200 in romanesque architectural style. The aisle had two bays; the choir was rectangular. The height of the tower was 15 to 20 meters by estimation. In the early modern period, people tore down the church in order to pave the street passing by. In 2006, the local historical society reconstructed the foundation walls of the church.
David Bergemann, Jan Novacek, Hans-Georg Stephan, Stefanie Thews: Archäologie interdisziplinär – Dorfwüstung Winnefeld im Solling In: Archäologie in Niedersachsen. 8, 2005, p 121–124. ISSN 1615-7265
Fort Albany was a bastioned earthwork that the Union Army built in Arlington County in Virginia. The Army constructed the fort during May 1861 as part of its Civil War defenses of Washington; the fort had a perimeter of 429 emplacements for 12 guns. Fort Richardson, Fort Craig and Fort Tillinghast provided supporting fire for the fort. A May 17, 1864, report from the Union Army's Inspector of Artillery noted the following:Fort Albany, Captain Rhodes commanding.–Garrison, one company First Massachusetts Volunteers–5 commissioned officers, 1 ordnance-sergeant, 145 men. Armament, two 24-pounder field howitzers, four 24-pounder siege, two Parrotts, one Coehorn mortar, one 10-inch mortar. Magazines, two. Ammunition, full supply and in good condition. Implements, complete and in good order. Drill in artillery, fair. Drill in infantry, fair. Discipline, fair. Garrison of sufficient strength. No trace of the fort remains, although a historic marker shows the location where the fort once stood, guarding the approach to the Long Bridge along the Columbia Turnpike, near the modern-day Pentagon.
The ground on which the Fort stood was cut away during the construction of the Henry G. Shirley Memorial Highway, in 1942. Drawing of Fort Albany in the collection of the Library of Congress Retrieved 2018-03-14
Masaaki Sakai is a popular Japanese performer from Tokyo. He is best known to English-speaking audiences as the title star of the TV show Monkey. Born the son of Shunji Sakai, a famous comedian in Japan, Sakai came to fame by fronting the group sounds band The Spiders; this group, formed in 1962, was popular throughout the 1960s. He took the title role of Son Goku in the 1970s Japanese TV program Saiyūki; this gained him fame in many English-speaking countries in the early 1980s when it was dubbed by the BBC and retitled Monkey. Due to his fame playing the mythical character Songoku, Sakai created a dance called "the Monkey" which became a craze in Japan. Sakai went on to a successful solo career after The Spiders disbanded, continued acting in films and on television. In 1999, he formed the band Sans Filtre with two former Spiders, Hiroshi Kamayatsu and Takayuki Inoue, they released their first album Yei Yei in 2000. Sakai is known in Japan as a car enthusiast and took part in the annual Mille Miglia race in Italy with his wife as a co-driver.
He won a similar Japanese road race on October 18, 2000 driving a 1947 Cisitalia 202 MM with Inoue Takayuki as co-driver. Due to business commitments he gave up racing in 2002 and gave his Alfa Romeo race car to Masahiko Kondō, a singer and race enthusiast, he is a hobby archer. Sakai has been married and divorced twice, has two daughters, he is an active supporter of AIDS charities. His show-business nickname is "Machaaki". Sakai starred in a series about travelling around Europe as well as the following series: Saiyūki Hakkutsu! Aruaru Daijiten Chubaw Desu yo! from 1994 to the present, where Sakai and a guest attempt to cook the dishes of famous restaurants. Shinano no Columbo
Old Shorter Hill is a summit in Rome, Georgia. With an elevation of 676 feet, Old Shorter Hill is the 910th highest summit in the US state of Georgia, it is considered to be one of the Seven Hills of Georgia. The hill was known as Shelton Hill, named after the main landowner in the area; the Cherokee Baptist Female College was housed in facilities located on the hill when it was founded in 1873. In 1877, Cherokee Baptist Female College was renamed Shorter College, thanks to generous donations from Alfred and Martha Shorter. In 1910, the college moved from Shelton Hill to its current location outside of downtown Rome; the hill's name was changed to Old Shorter Hill upon the death of Martha Shorter. There is a cul-de-sac on the summit