Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A deformation retraction of \(X\) onto \(A\) is a homotopy \(F\colon X\cross I\to X\), written \(F_t \coloneqq F(\wait, t)\), such that for all \(t\in I\): \[ F_0 = \id_X, \qquad F_1(X) \subseteq A, \qquad F_t(a) = a \text{ for all } a\in A .\] The subspace \(A\) is a deformation retract of \(X\) if there exists a deformation retraction of \(X\) onto \(A\).
If \(F\) is a deformation retraction of \(X\) onto \(A\), then \(r\coloneqq F_1\colon X\to A\) is a retraction, \(r\circ\iota = \id_A\), and \(F\) is a homotopy from \(\id_X\) to \(\iota\circ r\). Hence \(\iota\) is a homotopy equivalence with homotopy inverse \(r\).
Two spaces \(X\) and \(Y\) are homotopy equivalent if and only if there exists a space \(Z\) containing both \(X\) and \(Y\) as deformation retracts; for a homotopy equivalence \(f\colon X\to Y\), the mapping cylinder of \(f\) is such a \(Z\).
[@Hat02].