---
tag: D42X0F1EG1
kind: concept
name: Homotopy extension property
msc: [55P05]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_extension_property
same_as:
  wikidata: Q9382514
  wikipedia: Homotopy extension property
---

## Summary

Let $A$ be a subspace of a space $X$.
The pair $(X, A)$ has the homotopy extension property when each [[homotopy|9YA6S6HD9V]] on $A$ extends to all of $X$, once its start on $X$ is given.
Pairs with this property let one move maps on a subspace without losing control of the whole space.
It holds for CW pairs and is the defining property of [[cofibrations|W5WS5EY8ZM]].

## Explanations

### L1 human

Suppose you have a map on a big space $X$ and you start to wiggle it on a small part $A$.
The homotopy extension property says that you can wiggle the map on the rest of $X$ in a way that matches.
For example, a disk with its boundary circle has this property: any deformation of a map on the circle spreads into the disk.
Badly attached subspaces, such as a limit point of a sequence of isolated points, can fail it.