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Contractible Space
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a nonempty topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any path-connected space ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Domain
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This definition is immediately generalizable to any real, or complex, vector space. Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s_0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set. Definition Given two points x and y in a vector space X (such as Euclidean space \R^n), the convex hull of \ is called the and it is denoted by \left , y\right~:=~ \left\ ~=~ x + (y - x) , 1 where z , 1:= \ for every vector z. A subset S of a vector space X is said to be s_0 \in S if for every s \in S, the closed interval \left _0, s\right\subseteq S. A set S is and is called a if there exists some point s_0 \in S such that S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Mazurkiewicz
Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). His students included Karol Borsuk, Bronisław Knaster, Kazimierz Kuratowski, Stanisław Saks, and Antoni Zygmund. For a time Mazurkiewicz was a professor at the University of Paris; however, he spent most of his career as a professor at the University of Warsaw. The Hahn-Mazurkiewicz theorem, a basic result on curves prompted by the phenomenon of space-filling curves, is named for Mazurkiewicz and Hans Hahn. His 1935 paper ''Sur l'existence des continus indécomposables'' is generally considered the most elegant piece of work in point-set topology. During the Polish–Soviet War (1919–21), Mazurkiewicz as early as 1919 broke the most common Russian cipher for the Polish General Staff's cryptological agency. Thanks to this, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karol Borsuk
Karol Borsuk (8 May 1905 – 24 January 1982) was a Polish mathematician. His main area of interest was topology. He made significant contributions to Shape theory (mathematics), shape theory, a term which he coined. He also obtained important results in functional analysis. He was a professor of mathematics at the University of Warsaw, a member of the Polish Academy of Sciences, the Polish Mathematical Society and a leading representative of the Warsaw School (mathematics), Warsaw School of Mathematics. Life and career Early life and education Borsuk was born in 1905 in Warsaw to father Marian, a surgeon, and mother Zofia (née Maciejewska). In 1923, he graduated from the Stanisław Staszic State Gymnasium in Warsaw. Between 1923–1927, he studied mathematics at the Faculty of Philosophy of the University of Warsaw. He received his master's degree and doctorate from Warsaw University in 1927 and 1930, respectively. His PhD thesis title was ''On the Subject of Topological Chara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if every point admits a neighbourhood basis consisting of open path connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Simply Connected
In mathematics, a locally simply connected space is a topological space that admits a Base (topology), basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally simply connected space which is not simply connected. The Hawaiian earring is a space which is neither locally simply connected nor simply connected. The cone (topology), cone on the Hawaiian earring is contractible space, contractible and therefore simply connected, but still not locally simply connected. All topological manifolds and CW complexes are locally simply connected. In fact, these satisfy the much stronger property of being locally contractible space, locally contractible. A strictly weaker condition is that of being semi-locally simply connected. Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds. References Properties of top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Base
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...s of x. Definitions Neighbourhood of a point or set An of a point (or subset) x in a topological space X is any open subset U of X that contains x. A is any subset N \subseteq X that contains open neighbourhood of x; explicitly, N is a neighbourhood of x in X if and only if there exists some open subset U with x \in U \subseteq N. Equivalently, a neighborhood of x is any set that contains x in its topological interior. Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Allen Hatcher
Allen Edward Hatcher (born October 23, 1944) is an American mathematician specializing in geometric topology. Biography Hatcher was born in Indianapolis, Indiana. After obtaining his Bachelor of Arts, B.A. and Bachelor of Music, B.Mus. from Oberlin College in 1966, he went for his graduate studies to Stanford University, where he received his Doctor of Philosophy, Ph.D. in 1971. His thesis, ''A K2 Obstruction for Pseudo-Isotopies'', was written under the supervision of Hans Samelson. Afterwards, Hatcher went to Princeton University, where he was an National Science Foundation, NSF postdoc for a year, then a lecturer for another year, and then Assistant Professor from 1973 to 1979. He was also a member of the Institute for Advanced Study in 1975–76 and 1979–80. Hatcher moved to the University of California, Los Angeles as an assistant professor in 1977. From 1983 he has been a professor at Cornell University; he is now a emeritus, professor emeritus. In 1978 Hatcher was an Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
