Толық шектілік: Қысқалықтың жалғасы

Generalization of compactness
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.

A metric space is totally bounded if and only if for every real number , there exists a finite collection of open balls of radius whose centers lie in M and whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every , there exists a finite cover such that the radius of each element of the cover is at most This is equivalent to the existence of a finite ε net. A metric space is said to be totally bounded if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is compact if and only if it is closed and 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: every discrete ball of radius or less is a singleton, and no finite union of singletons can cover an infinite set.

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.)

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.

Every compact set is totally bounded, whenever the concept is defined. Every totally bounded set is bounded. A subset of the real line, or more generally of finite dimensional Euclidean space, is totally bounded if and only if it is bounded. The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded (in the norm topology) if and only if the space has finite dimension. Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem. A metric space is separable if and only if it is homeomorphic to a totally bounded metric space. The closure of a totally bounded subset is again totally bounded.

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 is totally bounded if and only if is complete.