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" (where the meaning of "size" depends on the structure of the ambient space.)
The term precompact (or pre-compact) 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 formula_1 is totally bounded if and only if for every real number formula_2, there exists a finite collection of open balls in "M" of radius formula_3 whose union contains . Equivalently, the metric space "M" is totally bounded if and only if for every formula_4, there exists a finite cover such that the radius of each element of the cover is at most formula_3. This is equivalent to the existence of a finite ε-net. A metric space is said to be Cauchy-precompact if every sequence admits a Cauchy subsequence; in metric spaces, a set is Cauchy-precompact if and only if it is totally bounded.
Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded). The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.
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 . (In other words, replaces the "size" , and a subset is of size if its Cartesian square 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 (Cauchy) 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 (not necessarily Hausdorff nor complete!).
The general logical form of the definition is: a subset formula_6 of a space formula_7 is totally bounded if and only if, given any size formula_8 there exists a finite cover of formula_6 such that each element of formula_6 has size at most formula_11 formula_7 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 formula_13 of the identity, a subset formula_14 is called (') ' if and only if formula_15 
A subset formula_6 of a topological group formula_7 is (') ' if it satisfies any of the following equivalent conditions:
The term usually appears in the context of Hausdorff topological vector spaces. 
In that case, the following conditions are also all equivalent to formula_6 being (left) totally bounded:
The definition of is analogous: simply swap the order of the products.
Condition 4 implies any subset of formula_19 is totally bounded (in fact, compact; see above). If formula_7 is not Hausdorff then, for example, formula_21 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, definition 1(b) 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 the norm topology) in terms of weakly convergent sequences of functionals: if formula_7 is a separable Banach space, then formula_14 is precompact if and only if every weakly convergent sequence of functionals converges uniformly on formula_24