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This PR introduces a few lemmas on (pseudo)metric spaces.

  • any complete metric space is a retract of its metric space of Cauchy approximations;
  • constant Cauchy approximations are convergent;
  • any metric space is a retract of its convergent Cauchy approximations;
  • short maps from a pseudometric space to a metric space reflect similarity.

@malarbol malarbol requested review from fredrik-bakke and lowasser and removed request for lowasser October 21, 2025 20:53
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Nice work, and it's nice to see you're active again :)

@malarbol
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it's nice to see you're active again :)

thanks. I needed a little break after this summer.
This is the first split of #1507 (as discussed in #1458 (comment)). I hoped to get further down the notion of Cauchy completion but this is the best I could do.

There will be upcoming PRs for

  • pseudometric completions of (pseudo)metric spaces;
  • metric quotients of pseudometric spaces;
  • metric pseudocompletion of (pseudo)metric spaces (the metric quotient of the pseudometric completion).

@fredrik-bakke fredrik-bakke merged commit 9998d39 into UniMath:master Oct 21, 2025
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@fredrik-bakke
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Looking forward to it!

@malarbol malarbol deleted the lemmas-metric-spaces branch October 21, 2025 23:23
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2 participants