Totally bounded space on arvau

Wikipedia / 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 covereded by finitelyly 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 is totally bounded, there exists a finite collection of open balls whose centers lie in M and whose union contains. Equivalently, the metric space M, there exists a finite cover. This is equivalent to the existence of a finite ε-net. A metric space is totally bounded iff 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: or less is a singleton, and no finite union of singletons can cover an infinite set.

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

In a metric space, every compact set is totally bounded.
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.
More generally, if a metric space has the Heine-Borel property, then a subset is bounded if and only if it is totally 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.

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.

Like compact sets, the union of finitely many totally bounded sets is totally bounded.
Unlike compact sets, every subset of a totally bounded set is again totally bounded.
The continuous image of a compact set is compact. The uniformly continuous image of a precompact set is precompact.

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 logicalal form of the definition is totally bounded if and only if, given any there exists is then totally bounded if and only if it is totally bounded when considered as a subset of itself.

is called () of a topological group is () if it satisfies any of the following equivalent conditions:

(where the right hand side is the Minkowski sum). -small. For any given filter subbase of the identity element's neighborhood filter is of the identity and every countably infinite 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

is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below.

under the canonical quotient is the identity element).

The term usually appears in the context of Hausdorff topological vector spaces. being (left) totally bounded:

In the completion is compact. is a Cauchy filter.

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

is totally bounded (in fact, compact; see 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.

is precompact if and only if every weakly convergent sequence of functionals converges uniformly

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 is complete.