Definition:Contractible Space

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

• Equivalence of Definitions of Contractible Space

• Results about contractible spaces can be found here.