View source for Totally bounded space

{{Short description|Generalization of compactness}}
In [[topology]] and related branches of [[mathematics]], '''total-boundedness''' is a generalization of [[Compact space|compactness]] for circumstances in which a set is not necessarily [[Closed (topology)|closed]].  A totally bounded set can be [[cover (topology)|cover]]ed by [[finite set|finite]]ly many [[subset]]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]]. These definitions coincide for subsets of a [[complete metric space]], but not in general.

== In metric spaces ==

[[File:Unit square totally bounded space.png|thumb|alt=A unit square can be covered by finitely many discs of radius ε < 1/2, 1/3, 1/4|[0, 1]<sup>2</sup> is a totally bounded space because for every ''ε'' > 0, the unit square can be covered by finitely many open discs of radius ''ε''.]]

A [[metric space]] <math> (M,d) </math> is '''''totally bounded''''' if and only if for every real number <math>\varepsilon > 0</math>, there exists a finite collection of [[open ball]]s of radius <math>\varepsilon</math>  whose centers lie in ''M'' and whose union contains&nbsp;{{mvar|M}}. Equivalently, the metric space ''M'' is totally bounded if and only if for every <math> \varepsilon >0</math>, there exists a [[finite cover]] such that the radius of each element of the cover is at most <math>\varepsilon</math>. This is equivalent to the existence of a finite [[ε-net (metric spaces)|ε-net]].{{sfn|Sutherland|1975|p=139}}  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.<ref>{{cite web |title=Cauchy sequences, completeness, and a third formulation of compactness |url=https://people.math.harvard.edu/~elkies/M55a.02/pdflatex/top6.pdf |website=Harvard Mathematics Department}}</ref>

Each totally bounded space is [[Bounded set|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:{{sfn|Willard|2004|p=182}} every discrete ball of radius <math>\varepsilon = 1/2</math> 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 {{mvar|S}} of a [[uniform space]] {{mvar|X}} is totally bounded if and only if, for any [[entourage (topology)|entourage]] {{mvar|E}}, there exists a finite cover of {{mvar|S}} by subsets of {{mvar|X}} each of whose [[Cartesian square]]s is a subset of {{mvar|E}}.  (In other words, {{mvar|E}} replaces the "size" {{math|''ε''}}, and a subset is of size {{mvar|E}} if its Cartesian square is a subset of {{mvar|E}}.)<ref name=":0">{{Cite book|last=Willard|first=Stephen|url=http://hdl.handle.net/2027/mdp.49015000696204|title=General topology|publisher=Addison-Wesley|year=1970|editor-last=Loomis|editor-first=Lynn H.|location=Reading, Mass.|pages=262|hdl=2027/mdp.49015000696204 }}  C.f. definition 39.7 and lemma 39.8.</ref>

The definition can be extended still further, to any category of spaces with a notion of [[compactness (topology)|compactness]] and [[Cauchy completion]]: a space is totally bounded if and only if its (Cauchy) completion is compact.

== Examples and elementary properties ==
* Every [[compact set]] is totally bounded, whenever the concept is defined.{{clarify|What does it mean the concept is defined,|date=January 2025}}
* 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 set|bounded]].<ref name=":1">{{Cite book|last1=Kolmogorov|first1=A. N.|url=http://hdl.handle.net/2027/mdp.49015000680570|title=Elements of the theory of functions and functional analysis|last2=Fomin|first2=S. V.|publisher=Graylock Press|year=1957|volume=1|location=Rochester, N.Y.|pages=51–3|hdl=2027/mdp.49015000680570 |translator-last=Boron|translator-first=Leo F.|orig-year=1954}}</ref>{{sfn|Willard|2004|p=182}}
* 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 (linear algebra)|dimension]].
* Equicontinuous bounded functions on a compact set are precompact in the [[Uniform convergence|uniform topology]]; this is the [[Arzelà–Ascoli theorem]].
* A [[metric space]] is [[Separable space|separable]] if and only if it is [[Homeomorphism|homeomorphic]] to a totally bounded metric space.{{sfn | Willard | 2004 | p=182}}
* The closure of a totally bounded subset is again totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

=== Comparison with compact sets ===
In metric spaces, a set is compact if and only if it is complete and totally bounded;<ref name=":1" /> 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 [[Uniform continuity|''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 axiom|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).{{sfn|Narici|Beckenstein|2011|pp=47-66}}{{sfn|Narici|Beckenstein|2011|pp=55-56}}{{sfn|Narici|Beckenstein|2011|pp=55-66}}

The general [[Logic (symbolic)|logic]]al form of the [[definition]] is: a subset <math>S</math> of a space <math>X</math> is totally bounded if and only if, [[given any]] size <math>E,</math> [[there exist]]s a finite cover <math>\mathcal{O}</math> of <math>S</math> such that each element of <math>\mathcal{O}</math> has size at most <math>E.</math>  <math>X</math> 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 <math>U \subseteq X</math> of the identity, a subset <math>S \subseteq X</math> is called ('''{{em|left}}''') '''{{em|<math>U</math>-small}}''' if and only if <math>(- S) +  S \subseteq U.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}  
A subset <math>S</math> of a [[topological group]] <math>X</math> is ('''{{em|left}}''') '''{{em|totally bounded}}''' if it satisfies any of the following equivalent conditions:

<ol>
<li>{{em|Definition}}: For any neighborhood <math>U</math> of the identity <math>0,</math> there exist finitely many <math>x_1, \ldots, x_n \in X</math> such that <math display="inline">S \subseteq \bigcup_{j=1}^n \left(x_j + U\right) := \left(x_1 + U\right) + \cdots + \left(x_n + U\right).</math></li>
<li>For any neighborhood <math>U</math> of <math>0,</math> there exists a finite subset <math>F \subseteq X</math> such that <math>S \subseteq F + U</math> (where the right hand side is the [[Minkowski sum]] <math>F + U := \{ f + u : f \in F, u \in U \}</math>).</li>
<li>For any neighborhood <math>U</math> of <math>0,</math> there exist finitely many subsets <math>B_1, \ldots, B_n</math> of <math>X</math> such that <math>S \subseteq B_1 \cup \cdots \cup B_n</math> and each <math>B_j</math> is <math>U</math>-small.{{sfn|Narici|Beckenstein|2011|pp=47-66}}</li>
<li>For any given [[Filters in topology|filter subbase]] <math>\mathcal{B}</math> of the identity element's [[neighborhood filter]] <math>\mathcal{N}</math> (which consists of all neighborhoods of <math>0</math> in <math>X</math>) and for every <math>B \in \mathcal{B},</math> there exists a cover of <math>S</math> by finitely many <math>B</math>-small subsets of <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}</li>
<li><math>S</math> is '''{{em|Cauchy bounded}}''': for every neighborhood <math>U</math> of the identity and every [[Countable set|countably infinite]] subset <math>I</math> of <math>S,</math> there exist distinct <math>x, y \in I</math> such that <math>x - y \in U.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}} (If <math>S</math> is finite then this condition is [[Vacuous truth|satisfied vacuously]]).</li>
<li>Any of the following three sets satisfies (any of the above definitions of) being (left) totally bounded:
<ol type="a">
<li>The [[Closure (topology)|closure]] <math>\overline{S} = \operatorname{cl}_X S</math> of <math>S</math> in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=47-66}}
* This set being in the list means that the following characterization holds: <math>S</math> is (left) totally bounded if and only if <math>\operatorname{cl}_X S</math> is (left) totally bounded (according to any of the defining conditions mentioned above). The same characterization holds for the other sets listed below.</li>
<li>The image of <math>S</math> under the [[First Isomorphism Theorem|canonical quotient]] <math>X \to X / \overline{\{ 0 \}},</math> which is defined by <math>x \mapsto x + \overline{\{ 0 \}}</math> (where <math>0</math> is the identity element).</li>
<li>The sum <math>S + \operatorname{cl}_X \{ 0 \}.</math>{{sfn|Schaefer|Wolff|1999|pp=12-35}}</li>
</ol></li>
</ol>

The term '''{{em|pre-compact}}''' usually appears in the context of Hausdorff topological vector spaces.{{sfn|Schaefer|Wolff|1999|p=25}}{{sfn|Trèves|2006|p=53}} 
In that case, the following conditions are also all equivalent to <math>S</math> being (left) totally bounded:

<ol start=7>
<li>In the [[Complete topological vector space|completion]] <math>\widehat{X}</math> of <math>X,</math> the closure <math>\operatorname{cl}_{\widehat{X}} S</math> of <math>S</math> is compact.{{sfn|Schaefer|Wolff|1999|p=25}}{{sfn|Jarchow|1981|pp=56-73}}</li>
<li>Every ultrafilter on <math>S</math> is a [[Cauchy filter]].</li>
</ol>

The definition of '''{{em|right totally bounded}}''' is analogous: simply swap the order of the products.

Condition 4 implies any subset of <math>\operatorname{cl}_X \{ 0 \}</math> is totally bounded (in fact, compact; see {{Slink||Comparison with compact sets}} above).  If <math>X</math> is not Hausdorff then, for example, <math>\{ 0 \}</math> is a compact complete set that is not closed.{{sfn|Narici|Beckenstein|2011|pp=47-66}}

=== Topological vector spaces ===
{{See also|Topological vector spaces#Properties}}

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 space]]s; it dates to a 1935 paper of John von Neumann.<ref>{{Cite journal|last=von Neumann|first=John|date=1935|title=On Complete Topological Spaces|journal=Transactions of the American Mathematical Society|volume=37|issue=1|pages=1–20|doi=10.2307/1989693|issn=0002-9947|doi-access=free|jstor=1989693 }}</ref> <!-- Theorem 11 in the cited paper; i.e. total boundedness of closure -->

This definition has the appealing property that, in a [[locally convex space]] endowed with the [[Weak topology (polar topology)|weak topology]], the precompact sets are exactly the [[Bounded set (topological vector space)|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 <math>X</math> is a separable Banach space, then <math>S \subseteq X</math> is precompact if and only if every [[Weak topology|weakly convergent]] sequence of functionals converges [[Uniform convergence|uniformly]] on <math>S.</math><ref>{{Cite journal|last=Phillips|first=R. S.|date=1940|title=On Linear Transformations|journal=Annals of Mathematics|page=525}}</ref>

==== Interaction with convexity ====

<ul>
<li>The [[Balanced set|balanced hull]] of a totally bounded subset of a topological vector space is again totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}}{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>The [[Minkowski sum]] of two compact (totally bounded) sets is compact (resp. totally bounded).</li>
<li>In a locally convex (Hausdorff) space, the [[convex hull]] and the [[disked hull]] of a totally bounded set <math>K</math> is totally bounded if and only if <math>K</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=67-113}}</li>
</ul>

==See also==

* [[Compact space]]
* [[Locally compact space]]
* [[Measure of non-compactness]]
* [[Orthocompact space]]
* [[Paracompact space]]
* [[Relatively compact subspace]]

== References ==
{{reflist}}

==Bibliography==
{{refbegin}} 
* {{Jarchow Locally Convex Spaces}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}}
* {{cite book | last=Sutherland|first= W. A.|author-link= Wilson Sutherland | title=Introduction to metric and topological spaces| publisher=Oxford University Press | year=1975 | isbn=0-19-853161-3 | zbl=0304.54002 }}
* {{Trèves François Topological vector spaces, distributions and kernels}}
* {{cite book | last=Willard|first= Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}
{{refend}}

{{DEFAULTSORT:Totally Bounded Space}}
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