**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](https://doi.org/10.1007/978-1-4612-6290-9) (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](https://en.wikipedia.org/wiki/Uniform_space) (equivalently, if $X$ is [completely regular](https://en.wikipedia.org/wiki/Tychonoff_space)), 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](https://mathoverflow.net/q/30662/102946) is loosely related.



Relative compactness in topological spaces (reference request)
See the *Handbook of Analysis and its Foundations*, by Eric Schechter (Section 17.15).


