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 S of a space X is totally bounded if and only if,
given any size E,
there exists a finite cover \mathcal{O} of S such that each element of \mathcal{O} has size at most E. X 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 U \subseteq X of the identity, a subset S \subseteq X is called ('
) ' if and only if (- S) + S \subseteq U. A subset S of a
topological group X is ('
) ' if it satisfies any of the following equivalent conditions: : For any neighborhood U of the identity 0, there exist finitely many x_1, \ldots, x_n \in X such that S \subseteq \bigcup_{j=1}^n \left(x_j + U\right). For any neighborhood U of 0, there exists a finite subset F \subseteq X such that S \subseteq F + U (where the right hand side is the
Minkowski sum F + U := \{ f + u : f \in F, u \in U \}). For any neighborhood U of 0, there exist finitely many subsets B_1, \ldots, B_n of X such that S \subseteq B_1 \cup \cdots \cup B_n and each B_j is U-small. For any given
filter subbase \mathcal{B} of the identity element's
neighborhood filter \mathcal{N} (which consists of all neighborhoods of 0 in X) and for every B \in \mathcal{B}, there exists a cover of S by finitely many B-small subsets of X. S is '''''': for every neighborhood U of the identity and every
countably infinite subset I of S, there exist distinct x, y \in I such that x - y \in U. (If S is finite then this condition is
satisfied vacuously). Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded: The
closure \overline{S} = \operatorname{cl}_X S of S in X. • This set being in the list means that the following characterization holds: S is (left) totally bounded if and only if \operatorname{cl}_X S is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below. The image of S under the
canonical quotient X \to X / \overline{\{ 0 \}}, which is defined by x \mapsto x + \overline{\{ 0 \}} (where 0 is the identity element). The sum S + \operatorname{cl}_X \{ 0 \}. The term '''''' usually appears in the context of Hausdorff topological vector spaces. In that case, the following conditions are also all equivalent to S being (left) totally bounded: In the
completion \widehat{X} of X, the closure \operatorname{cl}_{\widehat{X}} S of S is compact. Every ultrafilter on S is a
Cauchy filter. The definition of '''''' is analogous: simply swap the order of the products. Condition 4 implies any subset of \operatorname{cl}_X \{ 0 \} is totally bounded (in fact, compact; see above). If X is not Hausdorff then, for example, \{ 0 \} 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(a) 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 X is a separable Banach space, then S \subseteq X is precompact if and only if every
weakly convergent sequence of functionals converges
uniformly on S.
Interaction with convexity The
balanced hull of a totally bounded subset of a topological vector space is again totally bounded. The
Minkowski sum of two compact (totally bounded) sets is compact (resp. totally bounded). In a locally convex (Hausdorff) space, the
convex hull and the
disked hull of a totally bounded set K is totally bounded if and only if K is complete. ==See also==