Let \(X\) be a compact Hausdorff space. Suppose that for each \(x \in X\), there is a neighborhood \(U\) of \(x\) and a positive integer \(k\) such that \(U\) can be imbedded in \(\mathbb{R}^k\). Show that \(X\) can be imbedded in \(\mathbb{R}^N\) for some positive integer \(N\).
Solution
For each \(x\in X\), choose a neighborhood \(U_x\) and an embedding \[ e_x:U_x\hookrightarrow\mathbb R^{k_x}. \] Since compact Hausdorff spaces are normal, shrink twice: choose open sets \[ x\in V_x\subset\overline{V_x}\subset W_x\subset\overline{W_x}\subset U_x. \] By compactness choose finitely many \(V_1,\dots,V_m\) covering \(X\), with corresponding \(W_i,U_i,e_i:U_i\to\mathbb R^{k_i}\).
By Urysohn’s lemma choose continuous functions \[ \phi_i:X\to[0,1] \] such that \(\phi_i=1\) on \(\overline{V_i}\) and \(\phi_i=0\) on \(X\setminus W_i\). Since \(\overline{W_i}\) is compact and lies in \(U_i\), the coordinate functions of \(e_i\) are bounded on \(\overline{W_i}\). Hence the map \[ F_i:X\to\mathbb R^{k_i}, \qquad F_i(x)= \begin{cases} \phi_i(x)e_i(x),&x\in U_i,\\ 0,&x\notin U_i, \end{cases} \] is continuous: near the boundary of \(U_i\), \(\phi_i\) is already zero.
Define \[ F:X\to\mathbb R^N, \qquad F(x)=\bigl(\phi_1(x),F_1(x),\dots,\phi_m(x),F_m(x)\bigr), \] where \[ N=\sum_{i=1}^m(k_i+1). \] We show \(F\) is injective. If \(x\ne y\), choose \(i\) with \(x\in V_i\), so \(\phi_i(x)=1\). If \(\phi_i(y)\ne1\), the scalar coordinate \(\phi_i\) separates them. If \(\phi_i(y)=1\), then \(y\in W_i\subset U_i\), and \[ F_i(x)=e_i(x),\qquad F_i(y)=e_i(y), \] which are distinct because \(e_i\) is injective.
Thus \(F\) is a continuous injection from compact \(X\) into Hausdorff \(\mathbb R^N\). It is therefore a homeomorphism onto its image. Hence \(X\) embeds in some finite-dimensional Euclidean space.