Consider the following conditions on a locally \(m\)-euclidean space \(X\):
\(X\) is compact Hausdorff.
\(X\) is an \(m\)-manifold.
\(X\) is metrizable.
\(X\) is normal.
\(X\) is Hausdorff.
Show that (i) \(\Rightarrow\) (ii) \(\Rightarrow\) (iii) \(\Rightarrow\) (iv) \(\Rightarrow\) (v).
Solution
We prove the implications successively.
\((i)\Rightarrow(ii)\). A compact Hausdorff locally \(m\)-euclidean space has a finite atlas. Each coordinate neighborhood has a countable basis, and the union of finitely many such bases is a countable basis for \(X\). Thus \(X\) is Hausdorff, second-countable, and locally \(m\)-euclidean, hence an \(m\)-manifold.
\((ii)\Rightarrow(iii)\). An \(m\)-manifold is Hausdorff and second-countable, hence regular (indeed locally compact Hausdorff spaces are regular). By the Urysohn metrization theorem, a regular second-countable space is metrizable.
\((iii)\Rightarrow(iv)\). Every metrizable space is normal.
\((iv)\Rightarrow(v)\). A normal space is \(T_1\) by definition, so two distinct points are disjoint closed sets, and normality separates them by disjoint open sets.
Hence \[ (i)\Longrightarrow(ii)\Longrightarrow(iii)\Longrightarrow(iv)\Longrightarrow(v). \]