Totally Bounded Space - Definitions in Other Contexts

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, ..., A_n of subsets of X, such that S is contained in the union of the family (in other words, the family is a finite cover of S), and such that each set A_i in the family is of size E (or less). 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. (One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it is totally bounded when considered as a subspace.)

The terms "space" and "size" here are vague, and they may be made precise in various ways:

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. (In other words, a "size" here is a positive real number, and a subset is of size E if its diameter is less than E.) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a₁, a₂, ..., a_n of X such that S is contained in the union of the n open balls of radius E around the points a_i.

A subset S of a topological vector space, or more generally topological abelian group, X is totally bounded if and only if, given any neighbourhood E of the identity (zero) element of X, there exists a finite cover of S by subsets of X each of which is a translate of a subset of E. (In other words, a "size" here is a neighbourhood of the identity element, and a subset is of size E if it is translate of a subset of E.) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a₁, a₂, ..., a_n of X such that S is contained in the union of the n translates of E by the points a_i.

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 a_iE in place of E + a_i. 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 Ea_i in place of E + a_i. (In other words, a "size" here is unambiguously a neighbourhood of the identity element, but there are two notions of whether a set is of a given size: a left notion based on left translation and a right notion based on right translation.)

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. (In other words, a "size" here is an entourage, and a subset is of size E if its Cartesian square is a subset of E.) Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A₁, A₂, ..., A_n of X such that S is contained in the union of the A_i and, whenever the elements x and y of X both belong to the same set A_i, then (x,y) belongs to E (so that x and y are close as measured by 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.