How can one prove that manifolds are regular?

Question

First, some clarification of the definition of a manifold that I'm using:

A manifold $M$ is a Hausdorff, locally Euclidean and second countable topological space.

Now, I am trying to prove that Manifolds are paracompact, and I have established most of the details for the proof from the link above after getting so far under my own steam. The only hole I have, however, is the assertion that manifolds are regular. From what I can infer, this comes from the properties of being locally path-connected and Hausdorff, but I cannot make the leap from those two properties to the required regularity to complete the proof.

Apologies for the probably quite elementary question; I'm an applied mathematician, and am aiming to be well-read in a range of mathematical topics for my own interest, and while I have a textbook I'm working through on the topic of differentiable manifolds, there's still a certain unfamiliarity with the methods employed in certain pure maths topics.

Perhaps even a hint would be best, as I really do like to attempt to grasp these things by myself as much as possible, but I really just can't seem to get this result out... Thank you in advance for whatever help you can give me.

Answer 1

Note that locally Euclidean and Hausdorff implies locally compact Hausdorff. So the following more general result will answer your question.

Every locally compact Hausdorff space is completely regular. A proof can be found here.

The main idea is that the locally compact Hausdorff spaces are precisely the spaces which admit a one-point (or "Alexandroff") Hausdorff compactification. Now compact Hausdorff spaces are normal, hence completely regular. Normality need not be inherited by an arbitrary subspace, but complete regularity is.

[Note: in general I am a fan of the convention that "compact" and "locally compact" include the Hausdorff condition. So as to be maximally transparent, I am -- clearly -- not imposing that convention here.]