Difference between revisions of "Totally Bounded Metric Spaces"

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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]].)
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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.
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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 ==
 
== In metric spaces ==
  
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 in ''M'' of radius <math>\varepsilon</math> 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 '''''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.<ref name=":0" />
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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 balls in ''M'' of radius <math>\varepsilon</math> 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.  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 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}}
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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.
  
 
=== Uniform (topological) spaces ===
 
=== 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}}  C.f. definition 39.7 and lemma 39.8.</ref>
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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 {{mvar|E}}, there exists a finite cover of {{mvar|S}} by subsets of {{mvar|X}} each of whose Cartesian squares 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}}.)
  
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.
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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 properties ==
 
== Examples and elementary properties ==
* Every [[compact set]] is totally bounded, whenever the concept is defined.
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* Every compact set is totally bounded, whenever the concept is defined.
 
* Every totally bounded set is bounded.   
 
* 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|last=Kolmogorov|first=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|translator-last=Boron|translator-first=Leo F.|orig-year=1954}}</ref>{{sfn|Willard|2004|p=182}}
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* 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 [[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]].
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* 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.
* Equicontinuous bounded functions on a compact set are precompact in the [[Uniform convergence|uniform topology]]; this is the [[Arzelà–Ascoli theorem]].
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* Equicontinuous bounded functions on a compact set are precompact in the 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}}
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* A metric space is separable if and only if it is homeomorphic to a totally bounded metric space.
* The closure of a totally bounded subset is again totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}}
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* The closure of a totally bounded subset is again totally bounded.
  
 
=== Comparison with compact sets ===
 
=== 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.
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In metric spaces, a set is compact if and only if it is complete and totally bounded; 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.   
 
* 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.
 
* 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.
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* The continuous image of a compact set is compact.  The ''uniformly'' continuous image of a precompact set is precompact.
  
 
== In topological groups==
 
== 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}}
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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 (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 of <math>S</math> such that each element of <math>S</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.
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The general logical 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 exists a finite cover of <math>S</math> such that each element of <math>S</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}}  
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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 ('''''left''''') '''''<math>U</math>-small''''' if and only if <math>(- S) +  S \subseteq U.</math>   
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:
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A subset <math>S</math> of a topological group <math>X</math> is ('''''left''''') '''''totally bounded''''' if it satisfies any of the following equivalent conditions:
  
 
<ol>
 
<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>
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<li>'''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>
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<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>
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<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.</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>
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<li>For any given 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></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>
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<li><math>S</math> is '''''Cauchy bounded''''': for every neighborhood <math>U</math> of the identity and every 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> (If <math>S</math> is finite then this condition is satisfied vacuously).</li>
 
<li>Any of the following three sets satisfy (any of the above definitions) of being (left) totally bounded:
 
<li>Any of the following three sets satisfy (any of the above definitions) of being (left) totally bounded:
 
<ol type="a">
 
<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}}
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<li>The closure <math>\overline{S} = \operatorname{cl}_X S</math> of <math>S</math> in <math>X.</math>
 
* 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>
 
* 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>
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<li>The image of <math>S</math> under the 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>
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<li>The sum <math>S + \operatorname{cl}_X \{ 0 \}.</math></li>
 
</ol></li>
 
</ol></li>
 
</ol>
 
</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}}
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The term '''''pre-compact''''' usually appears in the context of Hausdorff topological vector spaces.  
 
In that case, the following conditions are also all equivalent to <math>S</math> being (left) totally bounded:
 
In that case, the following conditions are also all equivalent to <math>S</math> being (left) totally bounded:
  
 
<ol start=7>
 
<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>
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<li>In the completion <math>\widehat{X}</math> of <math>X,</math> the closure <math>\operatorname{cl}_{\widehat{X}} S</math> of <math>S</math> is compact.
<li>Every ultrafilter on <math>S</math> is a [[Cauchy filter]].</li>
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<li>Every ultrafilter on <math>S</math> is a Cauchy filter.</li>
 
</ol>
 
</ol>
  
The definition of '''{{em|right totally bounded}}''' is analogous: simply swap the order of the products.
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The definition of '''''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}}
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Condition 4 implies any subset of <math>\operatorname{cl}_X \{ 0 \}</math> is totally bounded.  If <math>X</math> is not Hausdorff then, for example, <math>\{ 0 \}</math> is a compact complete set that is not closed.
  
 
=== Topological vector spaces ===
 
=== 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, definition 1(b) 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|url=https://www.jstor.org/stable/1989693|journal=Transactions of the American Mathematical Society|volume=37|issue=1|pages=1–20|doi=10.2307/1989693|issn=0002-9947|doi-access=free}}</ref>
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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.
  
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]].
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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 <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>
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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 weakly convergent sequence of functionals converges uniformly on <math>S.</math>
  
 
==== Interaction with convexity ====
 
==== Interaction with convexity ====
  
 
<ul>
 
<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>
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<li>The balanced hull of a totally bounded subset of a topological vector space is again totally bounded.</li>
<li>The [[Minkowski sum]] of two compact (totally bounded) sets is compact (resp. totally bounded).</li>
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<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>
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<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.</li>
 
</ul>
 
</ul>
  

Latest revision as of 20:18, 13 November 2021

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 spaces

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. 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 (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric 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 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.)

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 properties

  • Every compact set is totally bounded, whenever the concept is defined.
  • 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.
  • 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.
  • Equicontinuous bounded functions on a compact set are precompact in the uniform topology; this is the Arzelà–Ascoli theorem.
  • A metric space is separable if and only if it is homeomorphic 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 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 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 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.
  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. is Cauchy bounded: for every neighborhood of the identity and every countably infinite subset of there exist distinct such that (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
      • 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

The term pre-compact usually appears in the context of Hausdorff topological vector spaces. 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.
  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. If 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 spaces; it dates to a 1935 paper of John von Neumann.

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

Interaction with convexity

  • The balanced hull of a totally bounded subset of a topological vector space is again totally bounded.
  • 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.

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