Let \(X\) be a space such that \(X \times I\) is normal. Let \(A\) be a closed subspace of \(X\); let \(f: A \to Y\) be a continuous map, where \(Y\) is an open subspace of \(\mathbb{R}^n\). If \(f\) is homotopic to a map that is extendable to a continuous map \(h: X \to Y\), then \(f\) itself is extendable to a continuous map \(g: X \to Y\), such that \(g \simeq h\).
Solution
Goal: Prove the Homotopy Extension Property for \((X, A)\) into an open set \(Y \subseteq \mathbb{R}^n\) under the hypothesis that \(X \times I\) is normal.
1. Setup of the cylinder extension problem:Proof:
1.1. Let \(H: A \times I \to Y\) be a homotopy with \(H(a, 0) = f(a)\) and \(H(a, 1) = h(a)\) for all \(a \in A\).
1.2. Define the closed subspace \(T \subset X \times I\):\[T = (X \times \{1\}) \cup (A \times I).\]
1.3. Define \(F: T \to Y \subseteq \mathbb{R}^n\) by:\[F(x, 1) = h(x) \quad \text{for } x \in X,\]\[F(a, t) = H(a, t) \quad \text{for } (a, t) \in A \times I.\]
1.4. On the intersection \((X \times \{1\}) \cap (A \times I) = A \times \{1\}\), \(F(a, 1) = h(a) = H(a, 1)\), so \(F\) is well-defined and continuous by the Pasting Lemma.
2. Extension into \(\mathbb{R}^n\) via Tietze Extension Theorem:Proof:
2.1. Since \(X \times I\) is normal, \(T\) is closed in \(X \times I\), and each coordinate map of \(F\) maps into \(\mathbb{R}\), by the Tietze Extension Theorem there exists a continuous extension:\[\widetilde{F}: X \times I \to \mathbb{R}^n \quad \text{such that } \widetilde{F}|_T = F.\]
3. Deformation into the open subset \(Y\):Proof:
3.1. Since \(Y\) is open in \(\mathbb{R}^n\), the preimage \(U = \widetilde{F}^{-1}(Y)\) is an open subset of \(X \times I\) containing \(T\).
3.2. The disjoint closed sets \(T\) and \((X \times I) \setminus U\) can be separated in the normal space \(X \times I\).
3.3. By Urysohn’s Lemma applied on slices (or Tube Lemma along the compact fibers), there exists a continuous function \(\mu: X \to [0, 1]\) such that \(\mu(A) = \{0\}\) and \(\{ (x, t) \in X \times I \mid t \ge \mu(x) \} \subseteq U\).
3.4. Define \(K: X \times I \to Y\) by:\[K(x, t) = \widetilde{F}(x, \, (1-t)\mu(x) + t).\]
3.5. For every \((x, t) \in X \times I\), \((1-t)\mu(x) + t \ge \mu(x)\), so \((x, (1-t)\mu(x) + t) \in U\), meaning \(K(x, t) \in Y\).
3.6. Thus \(K: X \times I \to Y\) is continuous.
4. Verification of boundary values and extension:Proof:
4.1. For \(a \in A\), \(\mu(a) = 0\), so \(K(a, t) = \widetilde{F}(a, t) = H(a, t)\).
4.2. In particular, for \(t = 0\) and \(a \in A\):\[K(a, 0) = H(a, 0) = f(a).\]
4.3. For \(t = 1\) and all \(x \in X\):\[K(x, 1) = \widetilde{F}(x, 1) = h(x).\]
4.4. Define \(g: X \to Y\) by \(g(x) = K(x, 0)\).
4.5. Then \(g\) is continuous, \(g|_A = f\), and \(K\) is a homotopy from \(g\) to \(h\) in \(Y\).
5. Conclusion:\(f\) extends to a continuous map \(g: X \to Y\) such that \(g \simeq h\). Q.E.D.