Let \(X\) be a space that is locally \(m\)-euclidean. Show that \(X\) is metrizable if and only if \(X\) is paracompact Hausdorff.
Solution
1. If \(X\) is metrizable, then \(X\) is paracompact Hausdorff.
Distinct points of a metric space have disjoint open balls around them, so \(X\) is Hausdorff. Every metrizable space is paracompact by Stone’s theorem.
2. If \(X\) is paracompact Hausdorff, then \(X\) is metrizable.
Every point of \(X\) has a neighborhood homeomorphic to \(\RR^m\), which is metrizable, so \(X\) is locally metrizable. By the Smirnov metrization theorem, a paracompact Hausdorff space that is locally metrizable is metrizable.
3. Q.E.D.
Steps 1 and 2 give the two implications.