TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related branches of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, total-boundedness is a generalization of
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for circumstances in which a set is not necessarily closed. A totally bounded set can be
cover Cover or covers may refer to: Packaging, science and technology * A covering, usually - but not necessarily - one that completely closes the object ** Cover (philately), generic term for envelope or package ** Housing (engineering), an exterior ...
ed by
finite Finite is the opposite of Infinity, 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 ...
ly many
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of every fixed "size" (where the meaning of "size" depends on the structure of the
ambient space An ambient space or ambient configuration space is the space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, al ...
.) The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean
relatively compact In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. These definitions coincide for subsets of a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
, but not in general.

In metric spaces

A
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
$\left(M,d\right)$ is ''totally bounded'' if and only if for every real number $\varepsilon > 0$, there exists a finite collection of
open ball In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In mathematics, particularly topology, a cover of a Set (mathematics), set X is a collection of sets whose union includes X as a subset. Formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of sets U_\alpha, then C is a cove ...
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 ''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. 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 classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
(with the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

), but not in general. For example, an infinite set equipped with the
discrete metric Discrete in science is the opposite of continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random vari ...
is bounded but not totally bounded.

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 uniform is a type of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, securit ...
. A subset of a
uniform space In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and
Cauchy completion In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a limit of a sequence, limit that is also in or, alternatively, if every Cauchy sequence ...
: a space is totally bounded if and only if its (Cauchy) completion is compact.

Examples and elementary properties

* Every
compact set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
is totally bounded, whenever the concept is defined. * Every totally bounded set is bounded. * A subset of the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, or more generally of finite-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, is totally bounded if and only if it is bounded. * The
unit ball Unit may refer to: Arts and entertainment * UNIT 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) i ...
in a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, or more generally in a
Banach space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, is totally bounded (in the norm topology) if and only if the space has finite
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. * Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measur ...
. * A
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is separable if and only if it is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...
al form of the
definition A definition is a statement of the meaning of a term (a word In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language ...

is: a subset $S$ of a space $X$ is totally bounded if and only if,
given any In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
size $E,$ there exists 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 $\left(- S\right) + S \subseteq U.$ A subset $S$ of a
topological group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
$X$ 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 In geometry, the Minkowski sum (also known as Dilation (morphology), dilation) of two set (mathematics), sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : ...

$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 $\mathcal$ (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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 satisfy (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 $X \to X / \overline,$ 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 Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
.
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 (an Abstra ...
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 Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...
endowed with the
weak topology In mathematics, weak topology is an alternative term for certain initial topology, 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 initia ...
, 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

• The
balanced hull In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field (mathematics), field with an absolute value (algebra), absolute value function , \cdot , ) is a Set (mathematics), set such tha ...
of a totally bounded subset of a topological vector space is again totally bounded.
• The
Minkowski sum In geometry, the Minkowski sum (also known as Dilation (morphology), dilation) of two set (mathematics), sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : ...

of two compact (totally bounded) sets is compact (resp. totally bounded).
• In a locally convex (Hausdorff) space, the
convex hull In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

and the disked hull of a totally bounded set $K$ is totally bounded if and only if $K$ is complete.

*
Compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Locally compact space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
* 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 Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
*
Relatively compact subspace In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose topological closure, closure is compact space, compact. Properties Every subset of a compact topological ...

Bibliography

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