Topology Imbeddings of Manifolds Prove that every manifold

Topology - Imbeddings of Manifolds

Prove that every manifold is regular and hence memorizable. Where do you use the Hausdorff condition?

Solution

Let X be a space.

X is called regular if and only if for each point x in X and U a neighbourhood of x, there is an open set V such that

xVclVUxVclVU

Also we have that a manifold is locally Euclidean. (A topological manifold is a locally Euclidean Hausdorff space)

In mathematics, a completely metrizable space(metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on Xsuch that (X, d) is a complete metric space and d induces the topology

Thus here to prove that is it metrizable we use the Hausdorff space that distinct points have distinct neighbourhoods.