nLab
totally bounded space

Totally bounded spaces

Idea

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.

Definition

In the following definitions, ‘finite’ means Kuratowski-finite, or finitely indexed, for the purposes of constructive mathematics. All of these definitions are constructively equivalent.

The slickest definition for uniform spaces is probably this one:

Definition

A uniform space XX is totally bounded if every uniform cover of XX 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:

Definition

A uniform space XX is totally bounded if, for every entourage UU of XX, there is a finite open cover 𝒞\mathcal{C} of XX such that every set GG in 𝒞\mathcal{C} satisfies G×GUG \times G \subseteq U.

In fact, it is enough to consider only basic entourages for some base of the uniformity. Thus, we may specialise to gauge spaces:

Definition

A gauge space XX is totally bounded if, for every gauging distance dd of XX, there is a finite open cover 𝒞\mathcal{C} of XX such that every set in 𝒞\mathcal{C} has dd-diameter less than 11.

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 11 as above). Thus, we may specialise to metric spaces:

Definition

A metric space XX is totally bounded if, for every positive number ϵ\epsilon, there is a finite open cover 𝒞\mathcal{C} of XX such that every set in 𝒞\mathcal{C} has diameter less than ϵ\epsilon.

Properties

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.

Any product of totally bounded spaces is totally bounded. The totally bounded subspaces of a given space XX form an ideal in the power set of XX.

A subspace of a Cartesian space is totally bounded if and only if it is bounded. Every totally bounded metric space is separable.

Precompact spaces

Here is another definition of total boundedness, different in style from the others. It makes sense even more generally, for any Cauchy space:

Definition

A Cauchy space XX is precompact if its completion X¯\overline{X} 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 XX generates a uniform cover of X¯\overline{X}. 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.

Proximity spaces

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”.

References

  • HAF, Sections 19.14–20

Revised on August 28, 2012 02:40:29 by Mike Shulman (71.136.235.154)