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Motivation and context: For a subset $S$ of a metric space $(M,d)$, the following are two very classical compactness results in Analysis:

  • 1a) The set $S$ is compact if and only if each sequence in $S$ has a subsequence that converges to a point in $S$.

  • 1b) The set $S$ is relatively compact (i.e., has compact closure) in $M$ if and only if each sequence in $S$ has a subsequence that converges to a point in $M$.

Now consider the following analogous claims for a subset $S$ of a topological space $X$:

  • 2a) The set $S$ is compact if and only if each net in $S$ has a subnet that converges to a point in $S$.

  • 2b) The set $S$ is relatively compact in $X$ if and only if each net in $S$ has a subnet that converges to a point in $X$.

Assertion 2a) is also a classical result in point set topology. On the other hand, the implication "$\Leftarrow$" in 2b) does not hold, in general.

More precisely, the following holds:

  • (i) If $X$ is not Hausdorff, it may happen that $S$ is compact but not closed, and also has non-compact closure. This shows that 2b) fails, in general.

  • (ii) A bit more interestingly, 2b) can also fail in Hausdorff spaces. Indeed, a counterexample can be constructed if we chose $S$ to be an open half disc with one additional point, in the half-disc topology on the upper half plane; this topology is, for instance, described in Example 78 of Steen and Seebach's "Counterexamples in Topology (1978)". (It is not stated explicitly there that this space yields a counterexample for 2b), but that's not difficult to see.)

  • (iii) If $X$ is Hausdorff and the topology on $X$ is induced by a uniform structure (equivalently, if $X$ is completely regular), then 2b) does indeed hold.

Assertion (iii) is not extremely difficult to show, but it is not completely obvious, either. Moreover, (iii) is sometimes quite useful in operator theory. So for the sake of citation, the following question arises:

Question (reference request): Do you know a reference where (iii) is explicitly stated and proved?

Related question: This question is loosely related.

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  • $\begingroup$ Not sure if you will find it there, but I would take a look on the Handbook of Analysis and its Foundations, by Eric Schechter. $\endgroup$
    – Renan Mezabarba
    Commented Dec 17, 2020 at 15:36
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    $\begingroup$ Glad to hear that! I'll post it then :) $\endgroup$
    – Renan Mezabarba
    Commented Dec 17, 2020 at 16:42
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    $\begingroup$ @RenanManeliMezabarba: Thanks for posting the answer! By the way, this Handbook by Schechter is incredible! I've dreamed about such a book for years (but didn't know it really exists - until today). $\endgroup$
    – Jochen Glueck
    Commented Dec 17, 2020 at 17:02
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    $\begingroup$ I found this book when I was studying about some equivalences of the ultrafilter lemma, back in 2016. Since then it is my favorite book, and even an inspiration for my own book. $\endgroup$
    – Renan Mezabarba
    Commented Dec 17, 2020 at 17:16
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    $\begingroup$ Concerning the “counterexamples” in (i), I would argue that you use the wrong definition of "relatively compact". In a not necessarily Hausdorff space the natural definition is IMHO the following: $M\subseteq X$ is relatively compact in $X$ if there is a compact $K\subseteq X$ with $M\subseteq K$. With this definition, the “counterexamples” in (i) become empty. $\endgroup$
    – Martin Väth
    Commented Dec 24, 2020 at 12:00

2 Answers 2

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See the Handbook of Analysis and its Foundations, by Eric Schechter (Section 17.15).

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My guess is that even $T_3$ is already sufficient. I do not have access to the monograph Fletcher, Peter and Lindgren, William F., Quasi-uniform spaces, M. Dekker, New York, Basel 1982, in the moment, but this contains quite some results about completion and quasi-compactness. Perhaps it also contains results about the relation to relative compactness.

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  • $\begingroup$ Indeed, it is true in $T_3$-spaces. For instance, the Handbook of Analysis and its Foundations mentioned in Renan Maneli Mezabarba's answer states the result in $T_3$-spaces (in Section 17.15). I also had a look at the book Quasi-uniform spaces that you suggested: while I could not find exactly the same result there, the corollary to Proposition 3.15 seems to be at least related. Thank you for the reference! $\endgroup$
    – Jochen Glueck
    Commented Dec 27, 2020 at 21:52

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