Corollary. A space \(X\) can be imbedded as a closed subspace of \(\mathbb{R}^N\) for some \(N\) if and only if \(X\) is locally compact and Hausdorff with a countable basis, and has finite topological dimension.
Solution
1. Direct implication (\(\implies\)): Suppose \(h: X \to \mathbb{R}^N\) is an embedding onto a closed subspace \(h(X) \subseteq \mathbb{R}^N\).Proof:
1.1. Locally compact Hausdorff: \(\mathbb{R}^N\) is locally compact and Hausdorff.Closed subspaces of locally compact Hausdorff spaces are locally compact Hausdorff, so \(X \cong h(X)\) is locally compact Hausdorff.
1.2. Second-countable: \(\mathbb{R}^N\) is second-countable.Second-countability is hereditary to all subspaces, so \(X\) has a countable basis.
1.3. Finite topological dimension: By the dimension properties of Euclidean space, \(\dim \mathbb{R}^N = N\).Topological dimension is monotonic on subspaces, so \(\dim X = \dim h(X) \le N < \infty\).
2. Converse implication (\(\impliedby\)): Suppose \(X\) is locally compact Hausdorff, second-countable, with topological dimension \(\dim X = m < \infty\).Proof:
2.1. One-point compactification: Let \(X^* = X \cup \{\infty\}\) be the one-point compactification of \(X\).Since \(X\) is locally compact Hausdorff and second-countable, \(X^*\) is a compact metrizable space with \(\dim X^* = \dim X = m\).
2.2. Embedding of compactification: By the imbedding theorem for compact metrizable spaces (Theorem 50.5), a compact metrizable space of topological dimension \(m\) imbeds in \(\mathbb{R}^{2m+1}\), so there exists a topological embedding \(g: X^* \to \mathbb{R}^{2m+1}\).
2.3. Construction of closed embedding into \(\mathbb{R}^{2m+2}\): Let \(p_0 = g(\infty) \in \mathbb{R}^{2m+1}\).For \(x \in X\), \(g(x) \neq p_0\).Define \(F: X \to \mathbb{R}^{2m+1} \times \mathbb{R} = \mathbb{R}^{2m+2}\) by: \[F(x) = \left( g(x), \frac{1}{\|g(x) - p_0\|} \right).\]
2.4. Embedding and properness: - Since \(g\) and \(\|\cdot - p_0\|^{-1}\) are continuous, \(F\) is continuous.
Since \(g|_X\) is injective, \(F\) is injective.
If a sequence \((x_k)\) in \(X\) leaves every compact set of \(X\), then \(x_k \to \infty\) in \(X^*\).
Then \(g(x_k) \to g(\infty) = p_0\), so \(\|g(x_k) - p_0\| \to 0\), which forces the last coordinate \(\frac{1}{\|g(x_k) - p_0\|} \to \infty\).
Hence \(\|F(x_k)\| \to \infty\), showing that \(F\) is a proper map. <2>5. Any proper continuous injection into a Hausdorff space is a closed embedding, so \(F(X)\) is closed in \(\mathbb{R}^{2m+2}\).
3. Q.E.D.Step 1 proves the forward implication, and step 2 imbeds \(X\) as a closed subspace of \(\mathbb{R}^N\) with \(N = 2m+2\).