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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeAug 28th 2015

Stub for localic completion. I wonder to what extent this can be generalized beyond metric spaces; for uniform spaces or Cauchy spaces we don’t have a nice collection of canonical basis elements like the open balls.

• CommentRowNumber2.
• CommentAuthorspitters
• CommentTimeAug 28th 2015

For uniform spaces, do the references at uniform locale help?

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 28th 2015

Picado and Pultr’s book Frames and locales discusses completion of uniform locales in full generality, see VIII.6,7,8 and X there.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJan 9th 2016

I added the definition to localic completion.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 9th 2016

By the way, regarding #2 and #3 (which I never replied to): do any of those references talk about completion of a uniform space (not a uniform locale)?

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeJan 9th 2016

@MikeShulman: Much of Chapter VIII and X in Picado and Pultr’s book is devoted to the comparison between uniform spaces and uniform locales, and their notions of completion. In particular, completeness of uniform spaces is studied there under the name of Cauchy completeness.

• CommentRowNumber7.
• CommentAuthorMike Shulman
• CommentTimeJan 9th 2016

Do they consider a localic completion of a uniform space?