Totally bounded space

In topology and related branches of mathematics, a totally bounded space is a space that can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the given context). The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a totally bounded set, in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.

The term precompact (or pre-compact) is sometimes used with the same meaning, but pre-compact is also used to mean relatively compact. For subsets of a complete metric space these meanings coincide but in general they do not. See also use of the axiom of choice below.

Definition for a metric space

A metric space $(M,d)$ is totally bounded if and only if for every real number $\varepsilon >0$ , there exists a finite collection of open balls in M of radius $\varepsilon$ whose union contains M. Equivalently, the metric space M is totally bounded if and only if for every $\varepsilon >0$ , there exists a finite cover such that the radius of each element of the cover is at most $\varepsilon$ . This is equivalent to the existence of a finite ε-net.[1]

Each totally bounded space is bounded (as the union of finitely many bounded sets is bounded), but the converse is not true in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.

If M is Euclidean space and d is the Euclidean distance, then a subset (with the subspace topology) is totally bounded if and only if it is bounded.

A metric space is said to be Cauchy-precompact if every sequence admits a Cauchy subsequence. Note that Cauchy-precompact is not the same as precompact (relatively compact), because Cauchy-precompact is an intrinsic property of the space, while precompact depends on the ambient space. Thus for metric spaces we have: compactness = Cauchy-precompactness + completeness. It turns out that the space is Cauchy-precompact if and only if it is totally bounded. Therefore, both names (Cauchy-precompact and totally bounded) can be used interchangeably.

Definitions in other contexts

The general logical form of the definition is: a subset S of a space X is a totally bounded set if and only if, given any size E, there exist a natural number n and a family A₁, A₂, ..., A_n of subsets of X, such that S is contained in the union of the family (in other words, the family is a finite cover of S), and such that each set A_i in the family is of size E (or less). In mathematical symbols:

\forall E\;\exists n\in \mathbb {N} \;A_{1},A_{2},\ldots ,A_{n}\subseteq X\left(S\subseteq \bigcup _{i=1}^{n}A_{i}{\text{ and for }}i=1,\ldots ,n(\operatorname {size} (A_{i})\leq E)\right).\!

The space X is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself. (One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it is totally bounded when considered as a subspace.)

The terms "space" and "size" here are vague, and they may be made precise in various ways:

Metric spaces

A subset S of a metric space X is totally bounded if and only if, given any positive real number E, there exists a finite cover of S by subsets of X whose diameters are all less than E. (In other words, a "size" here is a positive real number, and a subset is of size E if its diameter is less than E.) Equivalently, S is totally bounded if and only if, given any E as before, there exist elements a₁, a₂, ..., a_n of X such that S is contained in the union of the n open balls of radius E around the points a_i.

Topological vector spaces

Throughout, X will be a topological abelian group whose operation will be denoted by addition (i.e. +) and whose identity element will be denoted by 0. Note that every topological vector space (TVS) is a topological abelian group.

Definition: If U is a neighborhood of 0 in X then we say that a subset S ⊆ X is U-small or small of order U if S - S ⊆ U (i.e. x - y ∈ U for all x, y ∈ S).[2]

Definition: A subset S of a topological abelian group X is totally bounded if it satisfies any of the following equivalent conditions:

for any neighborhood U of the identity (zero) element of X, there exists a finite subset F ⊆ X such that S ⊆ F + U;
- this definition of "totally bounded" was used by John von Neumann in 1935.
for any neighborhood U of the identity, there exist finitely many elements x₁, ..., x_n of X such that S is contained in the union of the n translates of U by the points a_i;
- in words, this means that given any neighborhood U of the identity, there exists a finite cover of S by subsets of X each of which is a translate of a subset of U.
- hence, a "size" here is a neighborhood of the identity element, and a subset is of size U if it is translate of a subset of U.
S is Cauchy bounded;
- S is called Cauchy bounded if for every neighborhood U of the identity element and every countably infinite subset I of S, there exist x, y ∈ I such that x ≠ y and x - y ∈ U.[2] (Note that if S is a finite set then this condition is satisfied vacuously).
for any neighborhood U of the identity, there exist finitely many subsets B₁, ..., B_n of X, each of which is U-small, such that S ⊆ B₁ ∪ ... ∪ B_n.[2]
for any subbase ℬ for the filter 𝒩 of all neighborhoods of 0 in X, for every B ∈ ℬ, there exists a cover of S by finitely many B-small subsets of X.[2]

and if in addition X is a Hausdorff space then we may add to this list:

the closure of S in the completion of X is compact;[3]

The term pre-compact is usually used only in the context of Hausdorff TVSs.[3][4] Note in particular that if X is a complete and Hausdorff TVS then a subset is pre-compact if and only if it is relatively compact (i.e. its closure in X is compact).

Properties

In any topological abelian group X:

A subset is compact if and only if complete and totally bounded.[2]
- In particular, if a subset of a complete topological abelian group X is closed and totally bounded then it is compact;[2] the converse holds if X is also Hausdorff.
The closure of a totally bounded subset is again totally bounded.[2]

In any TVS X:

Every totally bounded subset is bounded.[5]
The closure and the balanced bull of a totally bounded set is again totally bounded.[5][2]
Every finite subset is totally bounded.

In a locally convex Hausdorff TVS X:

The convex hull of a precompact set is again precompact.[6]
If X is also complete then, the closed convex hull of a compact subset is again compact.[6]

Topological groups

A topological group X is left-totally bounded if and only if it satisfies the definition for topological abelian groups above, using left translates. That is, use a_i + E in place of E + a_i. Alternatively, X is right-totally bounded if and only if it satisfies the definition for topological abelian groups above, using right translates. That is, use E + a_i in place of a_i + E. (In other words, a "size" here is unambiguously a neighbourhood of the identity element, but there are two notions of whether a set is of a given size: a left notion based on left translation and a right notion based on right translation.)

Uniform spaces

Generalising the above definitions, a subset S of a uniform space X is totally bounded if and only if, given any entourage E in X, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, a "size" here is an entourage, and a subset is of size E if its Cartesian square is a subset of E.) Equivalently, S is totally bounded if and only if, given any E as before, there exist subsets A₁, A₂, ..., A_n of X such that S is contained in the union of the A_i and, whenever the elements x and y of X both belong to the same set A_i, then (x,y) belongs to E (so that x and y are close as measured by 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 completion is compact.

Examples and nonexamples

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 Archimedean property is used.
The unit ball in a Hilbert space, or more generally in a Banach space, is totally bounded if and only if the space has finite dimension.
Every compact set is totally bounded, whenever the concept is defined.
Every totally bounded metric space is bounded. However, not every bounded metric space is totally bounded.[7]
A subset of a complete metric space is totally bounded if and only if it is relatively compact (meaning that its closure is compact).
In a locally convex space endowed with the weak topology the precompact sets are exactly the bounded sets.
A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.[7]
An infinite metric space with the discrete metric (the distance between any two distinct points is 1) is not totally bounded, even though it is bounded.

Relationships with compactness and completeness

There is a nice relationship between total boundedness and compactness:

Every compact metric space is totally bounded.

Every metric space that is complete (i.e. every Cauchy sequence of points in the space converges to a point within the space) and totally bounded is compact.

A uniform space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as a generalisation of the Heine–Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness (and also replace closedness with completeness).

There is a complementary relationship between total boundedness and the process of Cauchy completion: A uniform space is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded.)

Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This may be taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of precompactness, while still using a separate definition of total boundedness. Then it becomes a theorem that a space is totally bounded if and only if it is precompact. (Separating the definitions in this way is useful in the absence of the axiom of choice; see the next section.)

Use of the axiom of choice

The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundedness in elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true (that is, the proof does not require choice) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded. But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.

Notes

Sutherland 1975, p. 139.
Narici 2011, pp. 47-66.
Schaefer 1999, p. 25.
Treves 2006, p. 53.
Narici 2011, pp. 156-175.
Treves 2006, p. 67.
Willard 2004, p. 182.

References

Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
Sutherland, W.A. (1975). Introduction to metric and topological spaces. Oxford University Press. ISBN 0-19-853161-3. Zbl 0304.54002.

Trèves, François (August 6, 2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

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[FOOTNOTESutherland1975139-1] Sutherland 1975, p. 139.

[FOOTNOTENarici201147-66-2] Narici 2011, pp. 47-66.

[FOOTNOTESchaefer199925-3] Schaefer 1999, p. 25.

[FOOTNOTETreves200653-4] Treves 2006, p. 53.

[FOOTNOTENarici2011156-175-5] Narici 2011, pp. 156-175.

[FOOTNOTETreves200667-6] Treves 2006, p. 67.

[FOOTNOTEWillard2004182-7] Willard 2004, p. 182.

Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, Krein–Milman theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry, Riemann hypothesis
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem F. Riesz's theorem Hahn–Banach (hyperplane separation Vector-Valued Hahn-Banach) Open mapping (Banach–Schauder) (Bounded inverse) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous Linear maps Densely defined Homomorphism Functionals Norm Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled (Ultra-) Bornological Brauner Complete (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the Approximation property