Show that a retract of a contractible space is contractible.
Solution
Goal: Prove that if \(X\) is a contractible topological space and \(A \subseteq X\) is a retract of \(X\), then \(A\) is contractible.
1. Retraction and contractibility data:Proof:
1.1. Let \(i: A \hookrightarrow X\) be the canonical inclusion map.
1.2. Since \(A\) is a retract of \(X\), there exists a continuous map \(r: X \to A\) such that:\[r \circ i = \operatorname{id}_A.\]
1.3. Because \(X\) is contractible, the identity map \(\operatorname{id}_X\) is nullhomotopic: there exists a point \(x_0 \in X\) and a continuous homotopy \(H: X \times I \to X\) such that:\[H(x, 0) = x \quad \text{and} \quad H(x, 1) = x_0 \quad \text{for all } x \in X.\]
2. Construction of nullhomotopy for \(A\):Define \(K: A \times I \to A\) by:\[K(a, t) = r(H(i(a), t)).\]Proof:
2.1. The map \(K\) is continuous as the composition of continuous maps \(r \circ H \circ (i \times \operatorname{id}_I)\).
2.2. At \(t = 0\):\[K(a, 0) = r(H(i(a), 0)) = r(i(a)) = (r \circ i)(a) = a = \operatorname{id}_A(a).\]
2.3. At \(t = 1\):\[K(a, 1) = r(H(i(a), 1)) = r(x_0) \in A.\]
2.4. Setting \(a_0 = r(x_0) \in A\), \(K(a, 1) = a_0\) is constant for all \(a \in A\).
3. Conclusion:\(K\) is a continuous homotopy between the identity map \(\operatorname{id}_A\) and the constant map \(c_{a_0}: A \to A\).Thus \(\operatorname{id}_A\) is nullhomotopic, so \(A\) is contractible. Q.E.D.