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.

Relation with other properties

Stronger properties

• Homeomorphism
• Strong deformation retraction
• Homotopy retraction

Weaker properties

• Weak homotopy equivalence of topological spaces

Related notions

• Homotopy equivalence of chain complexes