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Add the category of uniform spaces - #312

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uniform-spaces
Aug 1, 2026
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Add the category of uniform spaces#312
ScriptRaccoon merged 1 commit into
mainfrom
uniform-spaces

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@ScriptRaccoon ScriptRaccoon commented Jul 27, 2026

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This PR adds the category of uniform spaces and uniformly continuous maps. Its properties have also been decided, with two exceptions (see below).

It turns out that there are many similarities with Top, but also some notable differences:

  • Unif is extensive, but not infinitary extensive, not even infinitary distributive.
  • Unif has no extremal cogenerating set. There is nothing analogous to the Sierpinski space in Unif.
  • Unif is co-Malcev (neither Top nor Met has this property).
  • Regular epimorphisms in Unif are harder to describe than expected.

Here, uniform spaces are not assumed to be separated. Later, we should add the category SepUnif of separated uniform spaces as well. (Unfortunately, many authors build separatedness into their definition of a uniform space, which causes unnecessary confusion.)

Undecided properties

Two properties have not yet been decided for Unif, namely regularity and coregularity. I believe that the answers are the same as for Top, namely that the category is coregular but not regular. Gemini has also provided proofs of these claims, but I have not yet had a chance to digest them and would like to merge this PR first. The proofs can be added in later PRs.

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ScriptRaccoon force-pushed the uniform-spaces branch 6 times, most recently from 74d0783 to 40d2677 Compare August 1, 2026 07:49
@ScriptRaccoon ScriptRaccoon linked an issue Aug 1, 2026 that may be closed by this pull request
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ScriptRaccoon marked this pull request as ready for review August 1, 2026 10:55
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ScriptRaccoon merged commit 0e3443d into main Aug 1, 2026
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ScriptRaccoon deleted the uniform-spaces branch August 1, 2026 11:06
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category of uniform spaces

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