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.

Stronger properties

 * Homeomorphism
 * Strong deformation retraction
 * Homotopy retraction

Weaker properties

 * Weak homotopy equivalence of topological spaces

Related notions

 * Homotopy equivalence of chain complexes