Deformation retract

Definition

Let $X$ be a topological space and let $A \subseteq X$ carry the subspace topology. The subspace $A$ is a deformation retract of $X$ if there is a homotopy $H \colon X \times [0, 1] \to X$ such that:

$H(x, 0) = x$ for all $x \in X$ ;
$H(x, 1) \in A$ for all $x \in X$ ;
$H(a, 1) = a$ for all $a \in A$ .

The subspace $A$ is a strong deformation retract of $X$ if moreover $H(a, t) = a$ for all $a \in A$ and $t \in [0, 1]$ .

Remarks

The map $r = H(\cdot, 1) \colon X \to A$ is a retraction, and the inclusion $i \colon A \to X$ is a homotopy equivalence with homotopy inverse $r$ . Some textbooks, such as Hatcher's, use the name deformation retraction for the strong version. Example: $S^{n-1}$ is a strong deformation retract of $\R^n \setminus \set{0}$ , via $H(x, t) = (1 - t) x + t x / \norm{x}$ ; clause 3 holds since $\norm{a} = 1$ for $a \in S^{n-1}$ . Non-example: a point $p$ of the circle $S^1$ is not a deformation retract of $S^1$ , since otherwise $S^1$ would be contractible.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Homotopy concept
Subspace topology concept
Deformation retract definition

Tag	`YW8HBW1GJX`
URI	`m2://YW8HBW1GJX`
Kind	definition
Writing	W0 stub
MSC	54C15, 55P10
Defines	Deformation retract
Uses	Homotopy, Subspace topology
Changed	2 November 2026

Deformation retract

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here