Contractible space

Remarks

Equivalently, $X$ is nonempty and the identity map $\id_X$ is null-homotopic. A homotopy from $\id_X$ to a constant map is called a contraction of $X$. The empty space is not contractible, since there is no map from a point to it. Example: a convex subset $C \neq \emptyset$ of $\R^n$ is contractible; for a fixed $c \in C$, the map $H(x, t) = (1 - t) x + t c$ stays in $C$ by convexity and contracts $C$ to $c$. Non-example: the circle $S^1$ is not contractible, since its fundamental group is $\Z$, not trivial. Every contractible space is path-connected.