Homotopy equivalence of topological spaces

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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

Weaker properties

Related notions