Prove the following.
Theorem. Let \(X\) be a locally compact Hausdorff space with a countable basis, such that every compact subspace of \(X\) has topological dimension at most \(m\). Then \(X\) is homeomorphic to a closed subspace of \(\mathbb{R}^{2m+1}\).
If \(f: X \to \mathbb{R}^N\) is a continuous map, we say \(f(x) \to \infty\) as \(x \to \infty\) if given \(n\), there is a compact subspace \(C\) of \(X\) such that \(f(x) > n\) for \(x \in X - C\).
Let \(\bar{\rho}\) be the uniform metric on \(\mathcal{C}(X, \mathbb{R}^N)\). Show that if \(f(x) \to \infty\) as \(x \to \infty\) and \(\bar{\rho}(f, g) < 1\), then \(g(x) \to \infty\) as \(x \to \infty\).
Show that if \(f(x) \to \infty\) as \(x \to \infty\), then \(f\) extends to a continuous map of one-point compactifications. Conclude that if \(f\) is injective as well, then \(f\) is a homeomorphism of \(X\) with a closed subspace of \(\mathbb{R}^N\).
Given \(f: X \to \mathbb{R}^N\) and given a compact subspace \(C\) of \(X\), let
\[ U_\epsilon(C) = \ts{f \mid \Delta(f \mid C) < \epsilon}. \]
Show that \(U_\epsilon(C)\) is open in \(\mathcal{C}(X, \mathbb{R}^N)\).
Show that if \(N = 2m + 1\), then \(U_\epsilon(C)\) is dense in \(\mathcal{C}(X, \mathbb{R}^N)\). [Hint: Given \(f\) and given \(\epsilon, \delta > 0\), choose \(g: C \to \mathbb{R}^N\) so that \(d(f(x), g(x)) < \delta\) for \(x \in C\), and \(\Delta(g) < \epsilon\). Extend \(f - g\) to \(h: X \to [-\delta, \delta]^N\) using the Tietze theorem.]
Show there exists a map \(f: X \to \mathbb{R}^N\) such that \(f(x) \to \infty\) as \(x \to \infty\). [Hint: Write \(X\) as the union of compact subspaces \(C_n\) such that \(C_n \subset \operatorname{Int} C_{n+1}\) for each \(n\).]
Let \(C_n\) be as in (e). Use the fact that \(\bigcap U_{1/n}(C_n)\) is dense in \(\mathcal{C}(X, \mathbb{R}^N)\) to complete the proof.
Solution
1. Part (a): Perturbation of proper maps.Proof:
1.1. Let \(n \in \mathbb{Z}_+\).Since \(f(x) \to \infty\), there exists a compact \(C \subset X\) such that \(\|f(x)\| > n + 1\) for all \(x \in X \setminus C\).
1.2. Since \(\bar{\rho}(f, g) < 1\), \(\|f(x) - g(x)\| \le 1\) for all \(x \in X\).
1.3. For any \(x \in X \setminus C\): \[\|g(x)\| \ge \|f(x)\| - \|f(x) - g(x)\| > (n + 1) - 1 = n.\]
1.4. Hence \(g(x) \to \infty\) as \(x \to \infty\).
2. Part (b): Continuous extension to one-point compactifications and closed embeddings.Proof:
2.1. Let \(X^* = X \cup \{\infty_X\}\) and \((\mathbb{R}^N)^* = \mathbb{R}^N \cup \{\infty\}\) be the one-point compactifications.
2.2. Define \(f^*: X^* \to (\mathbb{R}^N)^*\) by \(f^*(x) = f(x)\) for \(x \in X\) and \(f^*(\infty_X) = \infty\).
2.3. For any neighborhood \(V\) of \(\infty\) in \((\mathbb{R}^N)^*\), \(V\) contains \((\mathbb{R}^N \setminus \overline{B}_n) \cup \{\infty\}\).Since \(f(x) \to \infty\), there is a compact \(C \subset X\) with \(f(X \setminus C) \subseteq \mathbb{R}^N \setminus \overline{B}_n\), so \((f^*)^{-1}(V) \supseteq (X \setminus C) \cup \{\infty_X\}\), which is open in \(X^*\).Thus \(f^*\) is continuous.
2.4. If \(f\) is injective, then \(f^*\) is an injective continuous map from the compact Hausdorff space \(X^*\) into the Hausdorff space \((\mathbb{R}^N)^*\), hence a topological embedding.
2.5. The image \(f^*(X^*)\) is compact, hence closed in \((\mathbb{R}^N)^*\).
2.6. Restricting to \(X\), \(f(X) = f^*(X^*) \cap \mathbb{R}^N\) is closed in \(\mathbb{R}^N\), and \(f: X \to f(X)\) is a homeomorphism.
3. Part (c): Openness of \(U_\varepsilon(C)\).Proof:
3.1. The condition \(\Delta(f|_C) < \varepsilon\) is equivalent to saying that if \(x_1, x_2 \in C\) with \(d(x_1, x_2) \ge \varepsilon\), then \(f(x_1) \neq f(x_2)\).
3.2. The set \(S = \{(x_1, x_2) \in C \times C : d(x_1, x_2) \ge \varepsilon\}\) is compact.
3.3. If \(f \in U_\varepsilon(C)\), the continuous function \((x_1, x_2) \mapsto \|f(x_1) - f(x_2)\|\) is strictly positive on \(S\), so it achieves a minimum \(\delta = \min_S \|f(x_1) - f(x_2)\| > 0\).
3.4. If \(\bar{\rho}(f, g) < \frac{\delta}{3}\), then for all \((x_1, x_2) \in S\): \[\|g(x_1) - g(x_2)\| \ge \|f(x_1) - f(x_2)\| - 2\bar{\rho}(f, g) \ge \delta - \frac{2\delta}{3} = \frac{\delta}{3} > 0.\]
3.5. Thus \(g \in U_\varepsilon(C)\), so \(U_\varepsilon(C)\) is open.
4. Part (d): Density of \(U_\varepsilon(C)\) for \(N = 2m + 1\).Proof:
4.1. Since \(C\) is compact metrizable with \(\dim C \le m\), the maps with \(\Delta < \varepsilon\) are dense in \(\mathcal{C}(C, \mathbb{R}^{2m+1})\); this is the density lemma in the proof of the imbedding theorem for compact metrizable spaces of dimension \(m\).
4.2. Given \(f \in \mathcal{C}(X, \mathbb{R}^N)\) and \(\delta > 0\), choose \(g_0 \in \mathcal{C}(C, \mathbb{R}^N)\) such that \(\|f|_C - g_0\| < \delta\) and \(\Delta(g_0) < \varepsilon\).
4.3. By the Tietze Extension Theorem, extend the coordinate functions of \(g_0 - f|_C: C \to [-\delta, \delta]^N\) to a continuous map \(h: X \to [-\delta, \delta]^N\).
4.4. Define \(g = f + h \in \mathcal{C}(X, \mathbb{R}^N)\).Then \(\bar{\rho}(f, g) \le \delta\) and \(g|_C = g_0 \in U_\varepsilon(C)\), proving density.
5. Part (e): Existence of a proper map \(f_0: X \to \mathbb{R}^N\).Proof:
5.1. Since \(X\) is locally compact and second-countable, \(X = \bigcup_{n=1}^\infty C_n\) where each \(C_n\) is compact and \(C_n \subset \operatorname{Int} C_{n+1}\).
5.2. By Urysohn’s Lemma, choose continuous functions \(\phi_n: X \to [0, 1]\) with \(\phi_n(C_n) = \{1\}\) and \(\phi_n(X \setminus \operatorname{Int} C_{n+1}) = \{0\}\).
5.3. Define \(\psi: X \to \mathbb{R}\) by \(\psi(x) = \sum_{n=1}^\infty (1 - \phi_n(x))\).For \(x \in X \setminus C_k\), \(\psi(x) \ge k - 1\).
5.4. The map \(f_0(x) = (\psi(x), 0, \dots, 0) \in \mathbb{R}^N\) is continuous and satisfies \(f_0(x) \to \infty\) as \(x \to \infty\).
6. Part (f): an injective map \(f\) with \(f(x) \to \infty\) exists.Proof:
6.1. Since \(\mathbb{R}^N\) is complete, \((\mathcal{C}(X, \mathbb{R}^N), \bar{\rho})\) is a complete metric space, hence a Baire space.
6.2. Let \(B = \{g \in \mathcal{C}(X, \mathbb{R}^N) : \bar{\rho}(f_0, g) < 1\}\).\(B\) is a non-empty open subspace, hence a Baire space.
6.3. For each \(n \in \mathbb{Z}_+\), \(U_{1/n}(C_n) \cap B\) is open and dense in \(B\) by 3 and
4.
4.4. By the Baire category theorem, the intersection \(G = \bigcap_{n=1}^\infty U_{1/n}(C_n) \cap B\) is dense in \(B\), hence non-empty.
4.5. Choose \(f \in G\).Since \(f \in B\), \(f(x) \to \infty\) as \(x \to \infty\) by
1.
1.6. For any distinct \(x_1, x_2 \in X\), there exists \(n\) such that \(x_1, x_2 \in C_n\) and \(d(x_1, x_2) \ge \frac{1}{n}\).Since \(f \in U_{1/n}(C_n)\), \(f(x_1) \neq f(x_2)\).Thus \(f\) is injective.
1.7. By 2, \(f\) is a homeomorphism of \(X\) onto a closed subspace of \(\mathbb{R}^{2m+1}\).Q.E.D.