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.

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.[3]

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 S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, E replaces the "size" ε, and a subset is of size E if its Cartesian square is a subset of E.)[2]

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.

In metric spaces, a set is compact if and only if it is complete and totally bounded;[4] without the axiom of choice only the forward direction holds. Precompact sets share a number of properties with compact sets.

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!).[5][6][7]

The definition of right totally bounded is analogous: simply swap the order of the products.

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.[12]

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.