Totally bounded space


In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size”.
The term precompact is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.

In metric spaces

A metric space is totally bounded if and only if for every real number, there exists a finite collection of open balls of radius whose centers lie in M and whose union contains . Equivalently, the metric space M is totally bounded if and only if for every, there exists a finite cover such that the radius of each element of the cover is at most. This is equivalent to the existence of a finite ε-net. A metric space is totally bounded iff every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and totally bounded.
Each totally bounded space is bounded. The reverse is true for subsets of Euclidean space, but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded: every discrete ball of radius or less is a singleton, and no finite union of singletons can cover an infinite set.

Uniform (topological) spaces

A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a uniform structure. A subset of a uniform space is totally bounded if and only if, for any entourage, there exists a finite cover of by subsets of each of whose Cartesian squares is a subset of.
The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its completion is compact.

Examples and elementary properties

Comparison with compact sets

In metric spaces, a set is compact if and only if it is complete and totally bounded; without the axiom of choice, only the forward direction holds. Precompact sets share a number of properties with compact sets.

In topological groups

Although the notion of total boundedness is closely tied to metric spaces, the greater algebraic structure of topological groups allows one to trade away some separation properties. For example, in metric spaces, a set is compact if and only if complete and totally bounded. Under the definition below, the same holds for any topological vector space.
The general logical form of the definition is: a subset of a space is totally bounded if and only if, given any size there exists a finite cover of such that each element of has size at most is then totally bounded if and only if it is totally bounded when considered as a subset of itself.
We adopt the convention that, for any neighborhood of the identity, a subset is called ' if and only if
A subset of a topological group is ' if it satisfies any of the following equivalent conditions:

  1. : For any neighborhood of the identity there exist finitely many such that
  2. For any neighborhood of there exists a finite subset such that .
  3. For any neighborhood of there exist finitely many subsets of such that and each is -small.
  4. For any given filter subbase of the identity element's neighborhood filter and for every there exists a cover of by finitely many -small subsets of
  5. is ': for every neighborhood of the identity and every countably infinite subset of there exist distinct such that .
  6. Any of the following three sets satisfies being totally bounded:

    1. The closure of in
      • This set being in the list means that the following characterization holds: is totally bounded if and only if is totally bounded. The same characterization holds for the other sets listed below.
    2. The image of under the canonical quotient which is defined by .
    3. The sum
The term ' usually appears in the context of Hausdorff topological vector spaces.
In that case, the following conditions are also all equivalent to being totally bounded:

  1. In the completion of the closure of is compact.
  2. Every ultrafilter on is a Cauchy filter.
The definition of is analogous: simply swap the order of the products.
Condition 4 implies any subset of is totally bounded. If is not Hausdorff then, for example, is a compact complete set that is not closed.

Topological vector spaces

Any topological vector space is an abelian topological group under addition, so the above conditions apply. Historically, statement 6 was the first reformulation of total boundedness for topological vector spaces; it dates to a 1935 paper of John von Neumann.
This definition has the appealing property that, in a locally convex space endowed with the weak topology, the precompact sets are exactly the bounded sets.
For separable Banach spaces, there is a nice characterization of the precompact sets in terms of weakly convergent sequences of functionals: if is a separable Banach space, then is precompact if and only if every weakly convergent sequence of functionals converges uniformly on

Interaction with convexity