Let \(X\) be a topological space, \(A\subseteq X\) a subspace, and \(I = [0,1]\). The pair \((X, A)\) has the homotopy extension property if for every space \(Y\), every continuous map \(f_0\colon X\to Y\), and every homotopy \(g_t\colon A\to Y\) with \(g_0 = f_0|_A\), there is a homotopy \(f_t\colon X\to Y\) starting at \(f_0\) with \(f_t|_A = g_t\) for all \(t\in I\). Equivalently, every pair of continuous maps \(X\times\ts{0}\to Y\) and \(A\times I\to Y\) that agree on \(A\times\ts{0}\) extends to a continuous map \(X\times I\to Y\).
If \(A\) is closed in \(X\), then \((X, A)\) has the homotopy extension property if and only if \(X\times\ts{0}\cup A\times I\) is a retract of \(X\times I\).
Every CW pair \((X, A)\) has the homotopy extension property.
See [@Hat02].