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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

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeAug 3rd 2018

Wrote out the proof of the metrization theorem.

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

Is it immediately obvious how to write down a metric for $X$, given $f$? I don’t know. A first guess might be to define the distance $X(p, q)$ to be the Hausdorff metric between $f^{-1}(p)$ and $f^{-1}(q)$, but it’s not so clear that this metric topology coincides with the original topology on $X$. The proofs of metrizability I’ve seen use $f$ pretty indirectly, concluding that $X$ 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 $[0, 1]$ 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 $X$ as opposed to a whole class of spaces.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeApr 4th 2019

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