Definition:Contractible Space
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Definition
Let $X$ be a topological space.
Definition 1
$X$ is called contractible if and only if the identity map $\operatorname{id}_X$ is homotopic to a constant map $X \to X$.
Definition 2
$X$ is called contractible if and only if it is homotopy equivalent to a point.
Examples
Real Euclidean Space
The real Euclidean space $\R^n$ is a contractible space for all $n \in \Z_{>0}$.
Also see
- Results about contractible spaces can be found here.