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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related branches of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, total-boundedness is a generalization of
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
for circumstances in which a set is not necessarily
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. A totally bounded set can be
cover Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of copy ...
ed by
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
ly many
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s 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 In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
. These definitions coincide for subsets of a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, but not in general.


In metric spaces

A
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
(M,d) is ''totally bounded'' if and only if for every real number \varepsilon > 0, there exists a finite collection of
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defi ...
s in ''M'' of radius \varepsilon whose union contains . 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. 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
(with the subspace topology), but not in general. For example, an infinite set equipped with the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
is bounded but not totally bounded: every discrete ball of radius \varepsilon = 1/2 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 In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
is totally bounded if and only if, for any entourage , there exists a finite cover of by subsets of each of whose
Cartesian square In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A ...
s is a subset of . (In other words, replaces the "size" , and a subset is of size if its Cartesian square is a subset of .) C.f. definition 39.7 and lemma 39.8. The definition can be extended still further, to any category of spaces with a notion of
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
and
Cauchy completion In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
: a space is totally bounded if and only if its (Cauchy) completion is compact.


Examples and elementary properties

* Every
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
is totally bounded, whenever the concept is defined. * Every totally bounded set is bounded. * A subset of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
, or more generally of finite-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, is totally bounded if and only if it is bounded. * The
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, or more generally in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, is totally bounded (in the norm topology) if and only if the space has finite
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. * Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem. * A
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is separable if and only if it is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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 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
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
al form of the
definition A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
is: a subset S of a space X is totally bounded if and only if,
given any In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
size E,
there exist In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
s a finite cover of S such that each element of S has size at most E. X 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 U \subseteq X of the identity, a subset S \subseteq X is called () if and only if (- S) + S \subseteq U. A subset S of a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
X is () if it satisfies any of the following equivalent conditions:
  1. : For any neighborhood U of the identity 0, there exist finitely many x_1, \ldots, x_n \in X such that S \subseteq \bigcup_^n \left(x_j + U\right) := \left(x_1 + U\right) + \cdots + \left(x_n + U\right).
  2. For any neighborhood U of 0, there exists a finite subset F \subseteq X such that S \subseteq F + U (where the right hand side is the Minkowski sum F + U := \).
  3. For any neighborhood U of 0, there exist finitely many subsets B_1, \ldots, B_n of X such that S \subseteq B_1 \cup \cdots \cup B_n and each B_j is U-small.
  4. For any given filter subbase \mathcal of the identity element's
    neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
    \mathcal (which consists of all neighborhoods of 0 in X) and for every B \in \mathcal, there exists a cover of S by finitely many B-small subsets of X.
  5. S is : for every neighborhood U of the identity and every
    countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
    subset I of S, there exist distinct x, y \in I such that x - y \in U. (If S is finite then this condition is satisfied vacuously).
  6. Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded:
    1. The closure \overline = \operatorname_X S of S in X. * This set being in the list means that the following characterization holds: S is (left) totally bounded if and only if \operatorname_X S 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 S under the canonical quotient X \to X / \overline, which is defined by x \mapsto x + \overline (where 0 is the identity element).
    3. The sum S + \operatorname_X \.
The term usually appears in the context of Hausdorff topological vector spaces. In that case, the following conditions are also all equivalent to S being (left) totally bounded:
  1. In the completion \widehat of X, the closure \operatorname_ S of S is compact.
  2. Every ultrafilter on S is a
    Cauchy filter In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
    .
The definition of is analogous: simply swap the order of the products. Condition 4 implies any subset of \operatorname_X \ is totally bounded (in fact, compact; see above). If X is not Hausdorff then, for example, \ 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, definition 1(b) was the first reformulation of total boundedness for
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s; it dates to a 1935 paper of John von Neumann. This definition has the appealing property that, in a
locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
endowed with the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
, 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 X is a separable Banach space, then S \subseteq X is precompact if and only if every weakly convergent sequence of functionals converges uniformly on S.


Interaction with convexity


See also

*
Compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
*
Locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
* Measure of non-compactness * Orthocompact space *
Paracompact space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
* Relatively compact subspace


References


Bibliography

* * * * * * {{DEFAULTSORT:Totally Bounded Space Uniform spaces Metric geometry Topology Functional analysis Compactness (mathematics)