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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 26th 2014

    Created a stub for Urysohn metrization theorem.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 26th 2014

    I put a Topology ToC on the page.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeMay 26th 2014

    Crosslinked with Pavel Urysohn

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 3rd 2018

    Wrote out the proof of the metrization theorem.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeAug 3rd 2018
    • (edited Aug 3rd 2018)

    I was always curious if any of various metrization theorems (Urysohn, Nagata-Smirnov, Bing, Moore, etc.) have an application where it is not immediately obvious how to write down a metric without invoking these theorems. The same question was asked by Jesse Kass without success on MathOverflow: https://mathoverflow.net/questions/93713/what-is-a-good-application-of-urysohns-theorem

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 4th 2018

    That’s a very good question, Dmitri.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 11th 2018
    • (edited Aug 11th 2018)

    Re #5, a possible test case is:

    If XX is a Hausdorff space that admits a continuous surjection f:[0,1]Xf: [0, 1] \to X, then XX is metrizable.

    Is it immediately obvious how to write down a metric for XX, given ff? I don’t know. A first guess might be to define the distance X(p,q)X(p, q) to be the Hausdorff metric between f 1(p)f^{-1}(p) and f 1(q)f^{-1}(q), but it’s not so clear that this metric topology coincides with the original topology on XX. The proofs of metrizability I’ve seen use ff pretty indirectly, concluding that XX is compact Hausdorff and second-countable, whereupon Urysohn metrization kicks in.

    The application then is to the “easy” direction of the Hahn-Mazurkiewicz theorem, the “only if” direction of the statement that a space is the continuous image of [0,1][0, 1] iff it is compact, metrizable, connected, and locally connected. It’s not quite an answer to the MO question, since that asked for a single space XX as opposed to a whole class of spaces.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2019

    Changed “named by Urysohn” to “named after Urysohn”

    diff, v9, current

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