Totally bounded space


In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every fixed "size". The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.
The term precompact is sometimes used with the same meaning, but pre-compact is also used to mean relatively compact. For subsets of a complete metric space
these meanings coincide but in general they do not. See also [|use of the axiom of choice] below.

Definition for a metric space

A metric space is totally bounded
if and only if for every real number, there exists
a finite collection of open balls in M of radius whose union contains M. 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.
Each totally bounded space is bounded, but the converse is not true in general.
For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.
If M is Euclidean space and d is the Euclidean distance, then
a subset is totally bounded if and only if it is bounded.
A metric space is said to be Cauchy-precompact if every sequence admits a Cauchy subsequence. Note that Cauchy-precompact is not the same as precompact, because Cauchy-precompact is an intrinsic property of the space, while precompact depends on the ambient space. Thus for metric spaces we have: compactness = Cauchy-precompactness + completeness. It turns out that the space is Cauchy-precompact if and only if it is totally bounded. Therefore, both names can be used interchangeably.

Definitions in other contexts

The general logical form of the definition is: a subset S of a space X is a totally bounded set if and only if, given any size E, there exist a natural number n and a family A1, A2,..., An of subsets of X, such that S is contained in the union of the family, and such that each set Ai in the family is of size E. In mathematical symbols:
The space X is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself.
The terms "space" and "size" here are vague, and they may be made precise in various ways:

Metric spaces

A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E. Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a1, a2,..., an of X such that S is contained in the union of the n open balls of radius E around the points ai.

Topological vector spaces

Throughout, X will be a topological abelian group whose operation will be denoted by addition and whose identity element will be denoted by 0.
Note that every topological vector space is a topological abelian group.

  1. for any neighborhood U of the identity element of X, there exists a finite subset F ⊆ X such that S ⊆ F + U;
  2. for any neighborhood U of the identity, there exist finitely many elements x1,..., xn of X such that S is contained in the union of the n translates of U by the points ai;
    • in words, this means that given any neighborhood U of the identity, there exists a finite cover of S by subsets of X each of which is a translate of a subset of U.
    • hence, a "size" here is a neighborhood of the identity element, and a subset is of size U if it is translate of a subset of U.
  3. S is Cauchy bounded;
    • S is called Cauchy bounded if for every neighborhood U of the identity element and every countably infinite subset I of S, there exist x, y ∈ I such that x ≠ y and x - y ∈ U..
  4. for any neighborhood U of the identity, there exist finitely many subsets B1,..., Bn of X, each of which is U-small, such that S ⊆ B1 ∪... ∪ Bn.
  5. for any subbase ℬ for the filter ? of all neighborhoods of 0 in X, for every B ∈ ℬ, there exists a cover of S by finitely many B-small subsets of X.
and if in addition X is a Hausdorff space then we may add to this list:

  1. the closure of S in the completion of X is compact;
The term pre-compact is usually used only in the context of Hausdorff TVSs.
Note in particular that if X is a complete and Hausdorff TVS then a subset is pre-compact if and only if it is relatively compact.

Properties

In any topological abelian group X:
In any TVS X:
In a locally convex Hausdorff TVS X:

Topological groups

A topological group X is left-totally bounded if and only if it satisfies the definition for topological abelian groups above, using left translates. That is, use ai + E in place of E + ai.
Alternatively, X is right-totally bounded if and only if it satisfies the definition for topological abelian groups above, using right translates. That is, use E + ai in place of ai + E.

Uniform spaces

Generalising the above definitions, a subset S of a uniform space X is totally bounded if and only if, given any entourage E in X, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A1, A2,..., An of X such that S is contained in the union of the Ai and, whenever the elements x and y of X both belong to the same set Ai, then belongs to E.
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 nonexamples

There is a nice relationship between total boundedness and compactness:
Every compact metric space is totally bounded.
Every metric space that is complete and totally bounded is compact.
A uniform space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as a generalisation of the Heine–Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness.
There is a complementary relationship between total boundedness and the process of Cauchy completion: A uniform space is totally bounded if and only if its Cauchy completion is totally bounded.
Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This may be taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of precompactness, while still using a separate definition of total boundedness. Then it becomes a theorem that a space is totally bounded if and only if it is precompact.

Use of the axiom of choice

The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundedness in elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded. But it is no longer true that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.