Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a homotopy equivalence if there is a continuous map \(g\colon Y\to X\) with \(f\circ g\simeq\id_Y\) and \(g\circ f\simeq\id_X\), where \(\simeq\) denotes homotopy. Such a map \(g\) is a homotopy inverse of \(f\). The spaces \(X\) and \(Y\) are homotopy equivalent, or have the same homotopy type, written \(X\simeq Y\), if there is a homotopy equivalence \(X\to Y\).
Every homeomorphism is a homotopy equivalence, and the converse fails: for \(n\geq 1\) the inclusion \(\ts{0}\hookrightarrow\RR^n\) is a homotopy equivalence, with homotopy inverse the constant map \(\RR^n\to\ts{0}\) and homotopy \(H(x, t) = tx\) from the constant map to \(\id_{\RR^n}\), but \(\ts{0}\) and \(\RR^n\) are not homeomorphic.