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How paracompact is that?

Dominic on stan-users writes:

I was reading through and came across the term with which I was not familiar: “paracompact.” I wrote a short blog post about it: 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:

    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:

    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

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