Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(\iota\colon A\injects X\) the inclusion. A retraction of \(X\) onto \(A\) is a continuous map \(r\colon X\to A\) with \(r\circ\iota=\id_A\), that is, \(r(a)=a\) for every \(a\in A\):
The subspace \(A\) is a retract of \(X\) if a retraction of \(X\) onto \(A\) exists.
Let \(A\subseteq X\) be a subspace with inclusion \(\iota\colon A\injects X\).
Every retraction \(r\colon X\to A\) is surjective.
Continuous maps \(r\colon X\to A\) with \(r\circ\iota=\id_A\) correspond bijectively to continuous maps \(\rho\colon X\to X\) with \(\rho(X)=A\) and \(\rho(a)=a\) for all \(a\in A\), via \(\rho=\iota\circ r\).
If \(r\) is a retraction and \(a_0\in A\), then \(r_*\circ\iota_* = \id\) on \(\pi_1(A, a_0)\) and on \(H_n(A)\), so \(\iota_*\colon \pi_1(A, a_0)\to\pi_1(X, a_0)\) and \(\iota_*\colon H_n(A)\to H_n(X)\) are injective. For every point \(x_0\) of a topological space \(X\), the subspace \(\ts{x_0}\) is a retract of \(X\), via the constant map \(X\to\ts{x_0}\).