Let \(X\) be a space that is locally \(m\)-euclidean. Show that if \(X\) is metrizable, then each component of \(X\) is an \(m\)-manifold.
Solution
Let \(C\) be a component of the metrizable locally \(m\)-euclidean space \(X\).
A locally euclidean space is locally path connected, so its components are open. Hence \(C\) is itself locally \(m\)-euclidean and metrizable (therefore Hausdorff).
It remains to prove that \(C\) has a countable basis. The space \(C\) is locally compact, since every point has a neighborhood homeomorphic to \(\mathbb R^m\), and Hausdorff. It is metrizable, hence paracompact by Stone’s theorem. A connected space is its own unique component, so by E-GZX7B, applied to the locally compact paracompact Hausdorff space \(C\), the space \(C\) has a countable basis. Hence \(C\) is Hausdorff, second-countable, and locally \(m\)-euclidean: it is an \(m\)-manifold.