A space \(X\) is said to be locally \(m\)-euclidean if for each \(x \in X\), there is a neighborhood of \(x\) that is homeomorphic to an open set of \(\mathbb{R}^m\). Such a space \(X\) automatically satisfies the \(T_1\) axiom, but it need not be Hausdorff. However, if \(X\) is Hausdorff and has a countable basis, then \(X\) is called an \(m\)-manifold.
Throughout these exercises, let \(X\) be a space that is locally \(m\)-euclidean.
Show that \(X\) is locally compact and locally metrizable.
Solution
1. Let \(x\in X\). Since \(X\) is locally \(m\)-euclidean, choose an open neighborhood \(U\) of \(x\) and a homeomorphism\[\phi:U\to V,\]with \(V\subset\mathbb R^m\) open.
2. Local compactness at \(x\).
2.1. In \(\mathbb R^m\) there is \(r>0\) with\[\overline{B_{V}( \phi(x),r)}\subset V,\]where \(\overline{B_V}\) is closure in \(\mathbb R^m\).
2.2. Put\[K:=\phi^{-1}\!\left(\overline{B_{V}(\phi(x),r)}\right).\]Then \(K\subset U\) and \(K\) is compact because \(\phi^{-1}\) is a homeomorphism \(V\to U\) and closed balls in \(\mathbb R^m\) are compact.
2.3. Let\[W:=\phi^{-1}\!\left(B_V(\phi(x),r/2)\right).\]Since \(B_V(\phi(x),r/2)\) is open in \(V\), \(W\) is open in \(U\), hence open in \(X\), and \(x\in W\subset K\).
2.4. \(K\) is a compact neighborhood of \(x\) in \(X\), so \(X\) is locally compact at \(x\).
3. Local metrizability at \(x\).
3.1. Let \(d_V\) be the Euclidean metric restricted to \(V\).
3.2. Define for \(a,b\in U\)\[d_U(a,b):=d_V(\phi(a),\phi(b)).\]This is a metric on \(U\) and generates the subspace topology because \(\phi\) is a homeomorphism.
3.3. Since \(x\in U\), there is a metrizable neighborhood \(U\) of \(x\) in \(X\).
3.4. Therefore \(X\) is locally metrizable.