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Attention

This thread has been generalized to uniform spaces as general metric spaces.

Context

The context was the equivalence: $$K\text{ compact}\iff K\text{ totally bounded, complete}$$ That is a purely topological notion compared to a purely uniform notion and a hybrid notion.

Problem

Given a uniform structure $\mathcal{U}$.

Is there a way to define completeness intrinsically: $$\mathcal{U}\text{ complete}:\iff\ldots$$ The usual way is extrinsically via filters: $$\mathcal{U}\text{ complete}:\iff\left(\mathcal{F}:\mathcal{F}\text{ cauchy}\implies\mathcal{F}\text{ converges}\right)$$ (Note that filters are an order-theoretic notion!)

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  • $\begingroup$ completeness does depend on the metric, does it not? I'm not quite sure what you're asking here. Does totally bounded have a categorical definition? $\endgroup$ – Henno Brandsma Jun 16 '14 at 12:23
  • $\begingroup$ Yes completeness depends on the metric since cauchyness. Moreover one might ask (and I guess that was your question) wether two inequivalent metrics give rise to the same complete uniform space. Actually I'm not quite sure either but what I can say is that of course its not a categorical notion as the categorical product but it is a notion due to a specific category namely the category of metric spaces... $\endgroup$ – C-Star-W-Star Jun 16 '14 at 12:28
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    $\begingroup$ Which category is this -- in particular, which set of maps does it include? I can imagine several categories whose objects are all metric spaces. $\endgroup$ – Carl Mummert Jun 16 '14 at 12:38
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    $\begingroup$ Yes I thought about it too. I mean either it includes only distance preserving maps as morphisms or it does also include contraction or even more general Lipschitz continuous maps. I would say lets start for convenience with the distance preserving maps though I guess for issues like completeness or compactness the better choice would be lipschitz maps as morphisms... $\endgroup$ – C-Star-W-Star Jun 16 '14 at 12:44
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    $\begingroup$ You ask for a description of completeness then say you'd like one of compactness. Is this a mistake? Also, would you agree with replacing "the" with "a" as I've done in my latest edit here? :) $\endgroup$ – Shaun Jun 16 '14 at 12:57

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