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 spacesEdit

A metric space   is totally bounded if and only if for every real number  , there exists a finite collection of open balls in M of radius   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.[1] A metric space is said to be Cauchy-precompact if every sequence admits a Cauchy subsequence; in complete metric spaces, a set is Cauchy-precompact if and only if it is totally bounded.[2]

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] every discrete ball of radius   or less is a singleton, and no finite union of singletons can cover an infinite set.

Uniform (topological) spacesEdit

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.

Examples and elementary propertiesEdit

Comparison with compact setsEdit

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.

  • Like compact sets, a finite union of 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 groupsEdit

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 general logical form of the definition is: a subset   of a space   is totally bounded if and only if, given any size   there exists a finite cover of   such that each element of   has size at most     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   of the identity, a subset   is called (left)  -small if and only if  [5] A subset   of a topological group   is (left) totally bounded if it satisfies any of the following equivalent conditions:

  1. Definition: For any neighborhood   of the identity   there exist finitely many   such that  
  2. For any neighborhood   of   there exists a finite subset   such that   (where the right hand side is the Minkowski sum  ).
  3. For any neighborhood   of   there exist finitely many subsets   of   such that   and each   is  -small.[5]
  4. For any given filter subbase   of the identity element's neighborhood filter   (which consists of all neighborhoods of   in  ) and for every   there exists a cover of   by finitely many  -small subsets of  [5]
  5.   is Cauchy bounded: for every neighborhood   of the identity and every countably infinite subset   of   there exist distinct   such that  [5] (If   is finite then this condition is satisfied vacuously).
  6. Any of the following three sets satisfy (any of the above definitions) of being (left) totally bounded:
    1. The closure   of   in  [5]
      • This set being in the list means that the following characterization holds:   is (left) totally bounded if and only if   is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below.
    2. The image of   under the canonical quotient   which is defined by   (where   is the identity element).
    3. The sum  [8]

The term pre-compact usually appears in the context of Hausdorff topological vector spaces.[9][10] In that case, the following conditions are also all equivalent to   being (left) totally bounded:

  1. In the completion   of   the closure   of   is compact.[9][11]
  2. Every ultrafilter on   is a Cauchy filter.

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

Condition 4 implies any subset of   is totally bounded (in fact, compact; see § Comparison with compact sets above). If   is not Hausdorff then, for example,   is a compact complete set that is not closed.[5]

Topological vector spacesEdit

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.

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   is a separable Banach space, then   is precompact if and only if every weakly convergent sequence of functionals converges uniformly on  [13]

Interaction with convexityEdit

  • The balanced hull of a totally bounded subset of a topological vector space is again totally bounded.[5][14]
  • 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.[15]

See alsoEdit

ReferencesEdit

  1. ^ Sutherland 1975, p. 139.
  2. ^ a b Willard, Stephen (1970). Loomis, Lynn H. (ed.). General topology. Reading, Mass.: Addison-Wesley. p. 262. C.f. definition 39.7 and lemma 39.8.
  3. ^ a b c Willard 2004, p. 182.
  4. ^ a b Kolmogorov, A. N.; Fomin, S. V. (1957) [1954]. Elements of the theory of functions and functional analysis,. Vol. 1. Translated by Boron, Leo F. Rochester, N.Y.: Graylock Press. pp. 51–3.
  5. ^ a b c d e f g h i Narici & Beckenstein 2011, pp. 47–66.
  6. ^ Narici & Beckenstein 2011, pp. 55–56.
  7. ^ Narici & Beckenstein 2011, pp. 55–66.
  8. ^ Schaefer & Wolff 1999, pp. 12–35.
  9. ^ a b Schaefer & Wolff 1999, p. 25.
  10. ^ Trèves 2006, p. 53.
  11. ^ Jarchow 1981, pp. 56–73.
  12. ^ von Neumann, John (1935). "On Complete Topological Spaces". Transactions of the American Mathematical Society. 37 (1): 1–20. doi:10.2307/1989693. ISSN 0002-9947.
  13. ^ Phillips, R. S. (1940). "On Linear Transformations". Annals of Mathematics: 525.
  14. ^ Narici & Beckenstein 2011, pp. 156–175.
  15. ^ Narici & Beckenstein 2011, pp. 67–113.

BibliographyEdit