**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 *A*_{1}, *A*_{2}, ..., *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 *a*_{1}, *a*_{2}, ..., *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 *a*_{1}, *a*_{2}, ..., *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*_{i}*E* 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 *A*_{1}, *A*_{2}, ..., *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.

Read more about this topic: Totally Bounded Space

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