---
tag: R5WP8Z69Q8
kind: concept
name: Contractible space
msc: [55P10, 55M99]
sources:
  - url: https://en.wikipedia.org/wiki/Contractible_space
same_as:
  wikidata: Q230689
  wikipedia: Contractible space
---

## Summary

A contractible space is a topological space that has the [[homotopy type|SK0ZJGWXDP]] of a point.
Equivalently, it is nonempty and its identity map can be deformed continuously into a constant map.
All homotopy groups and all reduced homology groups of a contractible space vanish.
Contractible spaces are the trivial objects of homotopy theory.

## Explanations

### L1 human

A space is contractible when it can be shrunk continuously to a single point inside itself.
A disk, a ball, a tree and the whole space $\R^n$ are contractible.
For $\R^n$, slide every point $x$ to the origin along the segment $(1 - t) x$.
A circle is not contractible: shrinking it to a point would have to break it.