Show that if \(A\) is a deformation retract of \(X\), and \(B\) is a deformation retract of \(A\), then \(B\) is a deformation retract of \(X\).
Solution
Let \(H:X\times I\to X\) be a deformation retraction of \(X\) onto \(A\), and let \(K:A\times I\to A\) be a deformation retraction of \(A\) onto \(B\). Thus \[ H(x,0)=x,\quad H(x,1)\in A,\quad H(a,t)=a, \] and \[ K(a,0)=a,\quad K(a,1)\in B,\quad K(b,t)=b. \] Define \(F:X\times I\to X\) by \[ F(x,t)= \begin{cases} H(x,2t),&0\le t\le1/2,\\ K(H(x,1),2t-1),&1/2\le t\le1. \end{cases} \] The two formulas agree at \(t=1/2\), so the pasting lemma gives continuity. We have \(F(x,0)=x\) and \(F(x,1)\in B\). If \(b\in B\subset A\), then both \(H(b,t)=b\) and \(K(b,t)=b\), hence \(F(b,t)=b\) for all \(t\). Therefore \(F\) is a deformation retraction of \(X\) onto \(B\).