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Cannot really claim that I have immediate urgent motivation to study this question but it appeared to me long ago, I recalled it now by some reason and decided to ask it here.

There is a strong feeling that there must be some important topological property distinguishing some sort of "complete" topological spaces in the same way as complete metric spaces are distinguished among all metric spaces.

I have seen several conditions (like complete uniformizability or Čech completeness), and all of them presume or imply very strong separation properties. Is there any nice property which does not depend on separatedness?

To try and ask something more precise -

Is there a naturally formulable property of topological spaces such that (a) any uniformizable space with this property is completely uniformizable and any Tychonoff space with this property is Čech complete; (b) any compact space (however non-Hausdorff it might be) has this property.

I am aware that this is still very far from being precise, and most likely I overlooked some trap here, but still let me ask it as it is, maybe with some luck it can be made better with time...

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    $\begingroup$ It seems that the Cech completeness can be defined without separation axioms: a topological space $X$ is Cech complete if there exists a sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ such that for any filter $\mathcal F$ of closed subsets such that for every $n$ some set $F\in\mathcal F$ is contained in some set of $\mathcal U_n$ the intersection $\bigcap \mathcal F$ is compact and $\mathcal F$ tends to this intersection (in the natural sense). Is this good reformulations? $\endgroup$
    – Taras Banakh
    Commented Apr 17, 2016 at 22:06
  • $\begingroup$ @TarasBanakh Looks very attractive! Three questions: (1) could you say more precisely what do you mean by "tends in the natural sense"? (2) what about spaces with $({\mathcal U}_n)_n$ as above such that for no filter $\mathcal F$ of closed sets is some $F\in\mathcal F$ is contained in some set of ${\mathcal U}_n$ for every $n$? (3) the fact that the system of covers is countable looks somehow arbitrary, what happens if one asks for any system of covers, not necessarily countable? For example, the collection of all open covers? Or the single cover consisting of all opens? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 18, 2016 at 5:09
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    $\begingroup$ (1) Writing that $\mathcal F$ tends to $\bigcap\mathcal F$ I had in mind that for every neighborhood $U$ of $\bigcap \mathcal F$ some set $F\in\mathcal F$ is contained in $U$; (2) Such spaces should be quite exotic, in particular, not $T_1$; (3) You asked about Cech-complete spaces-- they are countable intersections of open subsets in compact spaces. That is why countable family is natural in this context. $\endgroup$
    – Taras Banakh
    Commented Apr 18, 2016 at 21:22
  • $\begingroup$ "Cannot really claim that I have immediate urgent motivation to study this question" -- hmmmm... $\endgroup$
    – Włodzimierz Holsztyński
    Commented May 2, 2017 at 1:01
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    $\begingroup$ @WłodzimierzHolsztyński -- hmmm? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 2, 2017 at 4:22

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