---
tag: CEK4SC20WK
kind: concept
name: Deformation retract
msc: [54C15, 55P10]
sources:
  - url: https://en.wikipedia.org/wiki/Retraction_(topology)
---

## Summary

A subspace $A$ of a space $X$ is a deformation retract when $X$ can be deformed continuously onto $A$.
At the end of the deformation, the points of $A$ are back where they started.
Then the inclusion of $A$ in $X$ is a [[homotopy equivalence|XZG6MTSWCN]], so $A$ and $X$ have the same homotopy invariants.
Deformation retractions are the main tool for replacing a space by a simpler one of the same [[homotopy type|SK0ZJGWXDP]].

## Explanations

### L1 human

Picture a thick object melting slowly onto a thin skeleton inside it, with the skeleton itself held in place.
At the end of the melting, everything lies on the skeleton.
For example, the plane without the origin melts onto the unit circle if each point moves along its ray towards the circle.
So the punctured plane and the circle have the same holes.