Definition of precompactness in a topological group $G$

Question

I have seen that the definition of precompact sets in a topological group $G$ is a bit tricky. Can someone please explain? I saw that it has to do something with totally bounded sets. Is there a more 'natural' definition of precompactness? For example, is it true to say that if I have a subset $K$ in $G$, such that the closure of $K$ is compact in $G$, then $K$ is a precompact set in $G$?

Answer 1

For example, is it true to say that if I have a subset $K$ in $G$, such that the closure of $K$ is compact in $G$, then $K$ is a precompact set in $G$?

Yes, that is true. Such sets are called "relatively compact (in $G$)", which is a slightly more restrictive condition than precompactness.

While precompactness is an intrinsic property of a uniform space - the most often encountered examples of uniform spaces are metric spaces and topological groups [and subsets of these] - relative compactness is an extrinsic property, it refers to the ambient space. But every relatively compact subset of a uniform space is precompact.

A uniform space $(X,\mathscr{U})$ is called precompact (or totally bounded) if for every entourage $V \in \mathscr{U}$, $X$ can be covered by finitely many sets which are "small of order $V$". A set $M$ is small of order $V$ if $M\times M \subset V$. This shows that every subspace of a precompact space is again precompact, and it is easily seen that every quasicompact uniform space is precompact. For metric spaces, we can use the entourages $A_\varepsilon = \{ (x,y) : d(x,y) \leqslant \varepsilon\}$, and then a set is "small of order $A_\varepsilon$" if and only if its diameter is $\leqslant \varepsilon$.

If we restrict our attention to Hausdorff spaces, we have the simple characterisation that a Hausdorff uniform space is precompact if and only if its completion is compact [and that a space is compact if and only if it is complete and precompact].

Is there a more 'natural' definition of precompactness?

For topological groups $G$, we can characterise precompactness as follows:

$S \subset G$ is precompact if and only if for every neighbourhood $U$ of $e$, there is a finite set $F_U$ such that $$S\subset \bigcup_{x\in F_U} x\cdot U.$$

I'm not sure whether that meets your expectations, but it seems a little easier to understand than the general formulation for uniform spaces.

And for subsets complete Hausdorff topological groups, the concepts of precompactness and relative compactness coincide. If a (Hausdorff) topological group is not complete, it has precompact subsets that are not relatively compact.