A space is totally bounded if it may be covered by finitely many sets of arbitrarily small size.
The Heine–Borel theorem, which states that a closed and bounded subset of the real line is compact (in the finite open subcover sense), applies to all cartesian spaces but not to general metric spaces. However, if we use two facts about the real line (which hold for all cartesian spaces) —that a subset is closed if and only if it is complete and that a subset is bounded if and only if it is totally bounded—, then we get a theorem that does apply to all metric spaces (at least assuming the axiom of choice): that a complete and totally bounded space is compact.
The concept (and the Heine–Borel theorem, in this sense) apply not only to metric spaces but to uniform spaces; like completeness, total boundedness is a uniform property.
The slickest definition for uniform spaces is probably this one:
A uniform space is totally bounded if every uniform cover of has a finite subcover.
Since uniform covers are not a common approach to uniform spaces, we unwrap the definition of uniform covers in terms of entouranges to get this definition:
A gauge space is totally bounded if, for every gauging distance of , there is a finite open cover of such that every set in has -diameter less than .
In fact, it is enough to consider only basic gauging distances for some base of the gauge, or even for some subbase of the gauge if we make the requirement for arbitrarily small diameters (rather than the fixed diameter as above). Thus, we may specialise to metric spaces:
A metric space is totally bounded if, for every positive number , there is a finite open cover of such that every set in has diameter less than .
All of these results hold constructively unless otherwise noted.
Every compact space is totally bounded; this is immediate from Definition 1, since every uniform cover is an open cover. Conversely, if one assumes the ultrafilter principle, then every complete space and totally bounded space is compact.
Here is another definition of total boundedness, different in style from the others. It makes sense even more generally, for any Cauchy space:
A Cauchy space is precompact if its completion is compact.
It is immediate that a Cauchy space is compact if and only if it is both complete and precompact.
Every precompact uniform space is totally bounded; using Definition 1, this may be proved by checking that any uniform cover of generates a uniform cover of . The converse, that every totally bounded space is precompact, is equivalent to the ultrafilter principle. Of course, many totally bounded spaces may be proved precompact on weaker assumptions; in particular, that a bounded subset of a cartesian space is precompact is equivalent to the fan theorem (and so also follows from the principle of excluded middle), a fact related to the Heine–Borel theorem.
The category of totally bounded uniform spaces and uniformly continuous functions is equivalent to the category of proximity spaces and proximally continuous functions. Thus, proximity spaces can be considered yet another axiomatization of “totally bounded space” that doesn’t rely on a pre-existing kind of “space”.