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@malarbol malarbol commented Oct 28, 2025

This PR introduces the following concepts:

  • metric extensions of pseudometric spaces: a metric space with an isometry from the pseudometric space into it;
  • complete metric extensions of pseudometric spaces: metric extensions where the target space is complete;
  • Cauchy dense metric extensions of pseudometric spaces: metric extensions where all the points of the target space are limits of images of Cauchy approximations in the pseudometric space;
  • Cauchy precompletions of pseudometric spaces: the metric quotient of the Cauchy pseudocompletion of the pseudometric space.

The Cauchy precompletion of a pseudometric space is Cauchy dense and any metric extension into a complete metric
space factors through its Cauchy precompletion.

malarbol and others added 30 commits October 21, 2025 22:40
…agda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…agda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…es.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
@malarbol malarbol changed the title Cauchy precompletions of (pseudo)metric spaces Cauchy precompletions of pseudometric spaces Nov 19, 2025
@malarbol malarbol marked this pull request as ready for review November 19, 2025 01:59
@malarbol malarbol requested a review from lowasser November 19, 2025 02:01
@malarbol
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This is my follow up on #1458. As I commented before, this Cauchy precompletion is the closest to a Cauchy completion I could get.
There's still room to define the Cauchy completion as a metric extension that is both complete and Cauchy dense and I believe any such metric space should be isometrically equivalent to the Cauchy precompletion, but I haven't worked it all yet.

(f : isometry-Pseudometric-Space A B)
where

map-isometry-cauchy-approximation-Pseudometric-Space :
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map-isometry-cauchy-approximation-Pseudometric-Space :
map-cauchy-approximation-isometry-Pseudometric-Space :

@malarbol
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Re-drafting this PR to split in and formalize a few results about metric extensions before talking about Cauchy precompletions. And maybe rework a bit the construction of metric-quotients-of-pseudometric-spaces: I think the construction there can be seen as the initial metric extension of a pseudometric space which corresponds to universal property I was looking for. Similarly, the Cauchy precompletion seems to play a special role among metric extensions of a given pseudometric space.

@malarbol malarbol marked this pull request as draft November 22, 2025 15:47
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2 participants