---
tag: 8T7XY797SV
kind: definition
name: Contractible space
msc: [55P10, 55M99]
defines: R5WP8Z69Q8
uses: [XZG6MTSWCN, SMEV5BSQ5S]
sources:
  - url: https://en.wikipedia.org/wiki/Contractible_space
---

## Definition

A topological space $X$ is *contractible* if the map from $X$ to a one-point space is a [[homotopy equivalence|XZG6MTSWCN]].

## Remarks

Equivalently, $X$ is nonempty and the identity map $\id_X$ is [[null-homotopic|SMEV5BSQ5S]].
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|S1XK3HXHFX]] is $\Z$, not trivial.
Every contractible space is [[path-connected|0WB6DH499T]].