Editing Totally bounded space (section)

== 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 }}  Cf. 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.