Contractible space - Wikipedia
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. See more
PropertiesA contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a … See more
See also• Fake 4-ball – Mathematical topological manifold See more
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Contractible space - Encyclopedia of Mathematics
WEBJul 1, 2020 · Contractible space. A topological space $X$ that is homotopy equivalent (see Homotopy type) to a one-point space; i.e., if there is a point $x \in X$ and a homotopy …
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Classifying space - Wikipedia
WEBIn mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of …
Simply connected space - Wikipedia
WEBA sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in …
What is the definition of contractible space? (It is not a duplicate)
WEBLet $X$ be a topological space and $x_0\in X$. If there is a continuous map $F:X\times[0,1]\rightarrow X$ such that $F(x,0)=x$ and $F(x,1)=x_0$ ( $x \in X$ ) and …
Contractible space - WikiMili, The Best Wikipedia Reader
WEBIn 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 …
Homology theory - Encyclopedia of Mathematics
WEBApr 25, 2022 · The homology groups are topological and also homotopy invariants: If $ f $ is a homotopy equivalence, then $ f _ {*} $ is an isomorphism. If $ X $ is a contractible …
general topology - Every simply connected space is contractible ...
WEBApr 26, 2018 · All the homotopy groups of a contractible space are trivial (because all the homotopy groups of the point you're contracting to are trivial). Consequently, if any $\pi_i …
algebraic topology - The definition of a contractible space ...
WEBA space having the homotopy type of a point is called contractible. So following this definition, I can only carried out the proof in such a way: We retract this space $X$ to $A …
Understanding Contractible Spaces: Path Connected vs Simply …
WEBApr 3, 2010 · In summary, a contractible space is homotopy equivalent to a single-point space {0}. As long as the space is path connected, it is contractible. The punctured disc …
Cohomology - Wikipedia
WEBIn mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological …
How do you show that $S^{\\infty}$ is contractible?
WEBOct 8, 2009 · I think... We seek to show that a homotopy from the identity map of S∞ ( idS∞) to a constant map can be constructed and thus it must be null-homotopic, i. e., …
Contractible Space/Examples - ProofWiki
WEBThe real Euclidean space Rn R n is a contractible space for all n ∈ Z>0 n ∈ Z > 0 . Category: Examples of Contractible Spaces.
Definition:Contractible Space - ProofWiki
WEBDefinition 1. X X is called contractible if and only if the identity map idX id X is homotopic to a constant map X → X X → X . Definition 2. X X is called contractible if and only if it is …
Is every contractible space a cone? - Mathematics Stack Exchange
WEBBut contractible finite spaces exist (there is even an entire relevant book, "Algebraic topology of finite topological spaces and applications", Barmak). One way to quickly …
Connected space - Wikipedia
WEBIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open …
Equivalence of Definitions of Contractible Space - ProofWiki
WEBThe following definitions of the concept of Contractible Space are equivalent: Definition 1 $X$ is called contractible if and only if the identity map $\operatorname{id}_X$ is …
general topology - A fiber bundle over Euclidean space is trivial ...
WEBJan 2, 2021 · Here is an easy proof of the fact that a smooth fiber bundle over a contractible base is trivial. Bott and Tu's book, mentioned in the original question, work in this …
Weakly contractible - Wikipedia
WEBIn mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial. Property. It follows from Whitehead's Theorem that if a CW-complex is …
Local Contractibility of Homogeneous, Locally Path-Connected …
WEBMar 26, 2021 · Is it possible for a compact, metric, topological group to be locally path-connected but not locally contractible? Here "locally contractible" means that every point …
the cone is contractible - Mathematics Stack Exchange
WEBLet $X$ be a topological space. I want to show that the cone $CX$ is contractible. Here we construct a deformation retraction from $CX$ to the tip point of the cone $$H_t: CX\to …
Artificial structures visible from space - Wikipedia
WEBArtificial structures visible from space without magnification include highways, dams, and cities. [1] [2] [3] The Great Wall of China, often cited as the only human-made structure …
Talk:Contractible space - Wikipedia
WEBIllustration of some contractible and non-contractible spaces. Spaces A, B, and C are contractible; spaces D, E, and F are not. The article lead states that, "Intuitively, a …
Space Cadet (film) - Wikipedia
WEBSpace Cadet is a 2024 American comedy film written and directed by Liz W. Garcia. It stars Emma Roberts, Tom Hopper, Poppy Liu, and Gabrielle Union. Premise. Rex Simpson is …