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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 26th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexCreated a stub for [[Urysohn metrization theorem]].

   Created a stub for Urysohn metrization theorem.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 26th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI put a Topology ToC on the page.

   I put a Topology ToC on the page.

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeMay 26th 2014
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexCrosslinked with [[Pavel Urysohn]]

   Crosslinked with Pavel Urysohn

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 3rd 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexWrote out the proof of the metrization theorem. <a href="https://ncatlab.org/nlab/revision/diff/Urysohn+metrization+theorem/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Urysohn+metrization+theorem/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Urysohn+metrization+theorem">current</a>

   Wrote out the proof of the metrization theorem.

   diff, v5, current

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 4th 2018
   • (edited Aug 4th 2018)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexI 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>

   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

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 4th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThat's a very good question, Dmitri.

   That’s a very good question, Dmitri.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 11th 2018
   • (edited Aug 11th 2018)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 4th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexChanged "named by Urysohn" to "named after Urysohn" <a href="https://ncatlab.org/nlab/revision/diff/Urysohn+metrization+theorem/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Urysohn+metrization+theorem/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Urysohn+metrization+theorem">current</a>

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

   diff, v9, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJan 31st 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded some references (there was none before) and more hyperlinks in the text <a href="https://ncatlab.org/nlab/revision/diff/Urysohn+metrization+theorem/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Urysohn+metrization+theorem/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Urysohn+metrization+theorem">current</a>

   added some references (there was none before) and more hyperlinks in the text

   diff, v12, current