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I'm studying manifolds (by myself) and I can't understand why in order for a manifold to be metrizable, it needs to be paracompact (that's what I have read). The definition of a manifold is:
Manifold is a topological space which is Hausdorff, second countable and locally homeomorphic to $\mathbb{R}^n$.
I'm trying to prove that it is metrizable (without paracompactness, unless the concept of paracompactness is hidden somewhere in the definition). My attempt (with the help of the internet) is:
Let $M$ be a manifold.
$\mathbb{R}^n$ is locally compact and since M is locally homeomorphic to it, then M is locally compact (topological invariance). $M$ is locally compact and Hausdorff and that means that there exists a one point compactification $M^*$ which is compact and Hausdorff. Every compact and Hausdorff space is regular, and so by Urysohn's theorem $M^*$ is metrizable. Every compact metric space is complete, and every open subset of it is a complete metric space. So $M$ is metrizable.
That means that paracompactness isn't needed or second countable is like paracompactness but stronger. Basically, the questions are:
Do we need paracompactness for metrizable manifolds?
Is paracompactness related to a space being second countable?
Is the proof correct?

mxaxc
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  • ummm. paracompact gives partitions of unity, which is how we get global things. Such as a Riemann metric that is smooth everywhere. ...... You have a book you are reading? – Will Jagy Mar 09 '22 at 03:51
  • I'm a physicist and I have taken a course in general relativity. I'm using my professors notes. (I've read books on topology such as Munkres and Topology without tears and I'm planning to start reading a book on differential geometry like Lee Introduction to manifolds) – mxaxc Mar 09 '22 at 06:00
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    Your definition of a manifold implies paracompactness, so there's no loss of generality in using it. – Kajelad Mar 09 '22 at 09:55
  • @Kajelad Thanks for your answer. – mxaxc Mar 09 '22 at 12:26
  • I like the chapter on the hyperbolic plane in Shifrin (user here on MSE) , https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf That is the crossover example, providing a coordinate patch with, in this case, a positive quadratic form on the tangent space; actually the tangent space is also written in base coordinates. Relativity uses an indefinite quadratic form... A bit more formal in Millman and Parker. https://kupdf.net/download/elements-of-differential-geometry-millman-parker_59f22620e2b6f5c4757d44e3_pdf – Will Jagy Mar 09 '22 at 16:23
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    As you mention Munkres (topology), I think you would like his manifolds book, Analysis on Manifolds https://fourier.math.uoc.gr/~papadim/calculus_on_manifolds/Munkres.pdf He gives, as a source, Spivak https://www.math.ncku.edu.tw/~rchen/2015%20Teaching/Calculus%20on%20manifolds%20-%20Spivak,%20M..pdf – Will Jagy Mar 09 '22 at 16:36
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  • Every metrisable space is paracompact and Hausdorff. 2) On the other hand, if $X$ is paracompact and Hausdorff, and every point of $X$ has a metrisable neighbourhood, then $X$ is metrisable. Thus a locally Euclidean Hausdorff space is metrisable iff it is paracompact. 3) A second-countable Hausdorff space is metrisable iff it is regular. 4) Since every paracompact Hausdorff space is normal Hausdorff and hence regular, it follows that a second-countable Hausdorff space is metrisable iff it is paracompact.
    • – Tyrone Mar 09 '22 at 16:55
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    @WillJagy I didn't know that Munkres had written a book on Manifolds. It looks good and I'm definitely going to check it out. – mxaxc Mar 09 '22 at 22:39