## How paracompact is that?

Dominic on stan-users writes:

I was reading through http://arxiv.org/pdf/1410.5110v1.pdf and came across the term with which I was not familiar: “paracompact.” I wrote a short blog post about it: https://idontgetoutmuch.wordpress.com/2016/04/17/every-manifold-is-paracompact. It may be of interest to other folks reading the aforementioned paper. I would have used a partition of unity to justify the corollary myself but now I understand paracompactness.

And Betancourt replied:

The relevance of paracompactness (and how it follows immediately from local compactness and second-countabilitity) is discussed in Lee’s “Smooth Manifolds.” That is by far the best reference on differential geometry that I have yet to come across.

I don’t know what they’re talking about but I thought it might interest some of you.

1. I never though I’d see this topic come up on this blog. I can’t resist chiming in.

Local compactness is nice, but one can get by without it. In one page, Mary Ellen Rudin proved Stone’s theorem that all metric spaces are paracompact:

http://www.ams.org/journals/proc/1969-020-02/S0002-9939-1969-0236876-3/S0002-9939-1969-0236876-3.pdf

For example, the Hilbert space L^2(R) is paracompact.

Speaking of infinite-dimensional spaces, a Banach manifold is paracompact if and only if it is metrizable:

http://vmm.math.uci.edu/PalaisPapers/LusternikSchnirelOnBanachMan.pdf

Banach manifolds apparently are useful for some proofs in the Calculus of variations.

2. Sade, that master of Functional Analysis.

3. Dzhaughn says:

+1 on Lee’s book