# Homotopy equivalence of topological spaces

## Definition

Suppose $A$ and $B$ are topological spaces. A homotopy equivalence between $A$ and $B$ is a map $f:A \to B$ such that there exists a map $g:B \to A$ for which $f \circ g$ is homotopic to the identity on $B$ and $g \circ f$ is homotopic to the identity on $B$.

Two topological spaces between which there exists a homotopy equivalence are termed homotopy-equivalent topological spaces.