|
Contractible Topological 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 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 space ''Y'', any two maps ''f'',''g'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitehead Manifold
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where he incorrectly claimed that no such manifold exists. A contractible manifold is one that can continuously be shrunk to a point inside the manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask whether ''all'' contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". In dimension 2, it follows, for example, from the Riemann mapping theorem. Dimension 3 presents the first counterexample: the Whitehead manifold. Construction Take a copy of S^3, the three-dimensional sphere. Now find a compact unknotted solid torus T_1 inside the sphere. (A solid torus is an ordinary three-dimensional doughnut, ... [...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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, 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 (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karol Borsuk
Karol Borsuk (May 8, 1905 – January 24, 1982) was a Polish mathematician. His main interest was topology, while he obtained significant results also in functional analysis. Borsuk introduced the theory of '' absolute retracts'' (ARs) and ''absolute neighborhood retracts'' (ANRs), and the cohomotopy groups, later called Borsuk– Spanier cohomotopy groups. He also founded shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century; in particular, his open problems stimulated the infinite-dimensional topology. Borsuk received his master's degree and doctorate from Warsaw University in 1927 and 1930, respectively; his PhD thesis advisor was Stefan Mazurkiewicz. He was a member of the Polish ... [...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 entirely of open, 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 space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Connected Space
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. 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 space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...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 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 on the Hawaiian earring is 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. A strictly weaker condition is that of being semi-locally simply connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comb Space
In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space. Formal definition Consider \R^2 with its standard topology and let ''K'' be the set \. The set ''C'' defined by: :(\ \times ,1) \cup (K \times ,1 \cup ( ,1\times \) considered as a subspace of \R^2 equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by: :\ \cup (K \times ,1 \cup ( ,1\times \) . This is the comb space with the line segment \ \times contractible,_but_not_locally_contractible,_Locally_connected_space.html" "title="Contractible_space.html" "title="locally_connected_space.html" "title=",1) deleted. Topological properties The comb space and the deleted comb space have some interesting topological prope ... [...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 neighbourhoods 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 also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.). There are many other types of neighbourhoods that are used i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
