---
tag: 6MJH22BCAE
kind: definition
name: Homotopy type
msc: [55P10, 55P15]
defines: SK0ZJGWXDP
uses: [XZG6MTSWCN, 0G8VEZ3WTK]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Definition

Two topological spaces $X$ and $Y$ are *homotopy equivalent*, or *have the same homotopy type*, written $X \simeq Y$, if there is a [[homotopy equivalence|XZG6MTSWCN]] $f \colon X \to Y$.
The *homotopy type* of $X$ is the class of all spaces homotopy equivalent to $X$.

## Remarks

The relation $\simeq$ is reflexive, symmetric and transitive, so it behaves as an [[equivalence relation|0G8VEZ3WTK]] on topological spaces.
Symmetry holds because a homotopy inverse is itself a homotopy equivalence.
Example: $\R^n$ has the homotopy type of a point; the maps $\R^n \to \set{0}$ and $\set{0} \to \R^n$ are mutually inverse up to the homotopy $H(x, t) = (1 - t) x$.
Non-example: $S^1$ and a point do not have the same homotopy type, since their [[fundamental groups|S1XK3HXHFX]] differ.
Homeomorphic spaces have the same homotopy type, but the converse fails, as $\R$ and a point show.