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-In [[topology]] and related branches of [[mathematics]], a '''totally bounded space''' is a space that can be [[cover (topology)|cover]]ed by [[finite set|finite]]ly many [[subset]]s of any fixed "size" (where the meaning of "size" depends on the given context). The smaller the size fixed, the more subsets may be needed, but any specific size should require only finitely many subsets. A related notion is a '''totally bounded set''', in which only a subset of the space needs to be covered. Every subset of a totally bounded space is a totally bounded set; but even if a space is not totally bounded, some of its subsets still will be.
+== right Longchamp Le Pliage Cuir Australia ==
-The term '''precompact''' (or '''pre-compact''') is sometimes used with the same meaning, but `pre-compact'  is also used to mean [[relatively compact]]. In a [[complete metric space]]
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-if and only if for every real number <math> \epsilon >0</math>, there exists
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-element of the cover is at most <math>\epsilon</math>. This is equivalent to the existence of a finite [[ε-net (metric spaces)|ε-net]].<ref>Sutherland p.139</ref>
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-Each totally bounded space is [[Bounded set|bounded]] (as the union of finitely many bounded sets is bounded), but the converse is not true in general.
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-a subset (with the [[subspace topology]]) is totally bounded if and only if it is bounded.
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-== Definitions in other contexts==
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-The general [[logic (symbolic)|logic]]al form of the [[definition]] is: A subset ''S'' of a space ''X'' is a totally bounded set if and only if, [[given any]] size ''E'', [[there exist]] a [[natural number]] ''n'' and a [[family of sets|family]] ''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub> of subsets of ''X'', such that ''S'' is contained in the [[union (set theory)|union]] of the family (in other words, the family is a ''finite cover'' of ''S''), and such that each set ''A''<sub>''i''</sub> in the family is of size ''E'' (or less). In [[mathematical symbols]]:
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-The space ''X'' is a totally bounded space if and only if it is a totally bounded set when considered as a subset of itself.
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-(One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it is totally bounded when considered as a [[subspace (topology)|subspace]].)
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-The terms "space" and "size" here are vague, and they may be made precise in various ways:
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-A subset ''S'' of a [[metric space]] ''X'' is totally bounded if and only if, given any [[positive number|positive]] [[real number]] ''E'', there exists a finite cover of ''S'' by subsets of ''X'' whose [[diameter]]s are all less than ''E''. (In other words, a "size" here is a positive real number, and a subset is of size ''E'' if its diameter is less than ''E''.) Equivalently, ''S'' is totally bounded if and only if, given any ''E'' as before, there exist elements ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> of ''X'' such that ''S'' is contained in the union of the ''n'' [[open ball]]s of radius ''E'' around the points ''a''<sub>''i''</sub>.
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-A subset ''S'' of a [[topological vector space]], or more generally [[topological abelian group]], ''X'' is totally bounded if and only if, given any [[neighbourhood (topology)|neighbourhood]] ''E'' of the [[identity element|identity (zero) element]] of ''X'', there exists a finite cover of ''S'' by subsets of ''X'' each of which is a [[translate (geometry)|translate]] of a subset of ''E''. (In other words, a "size" here is a neighbourhood of the identity element, and a subset is of size ''E'' if it is translate of a subset of ''E''.) Equivalently, ''S'' is totally bounded if and only if, given any ''E'' as before, there exist elements ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> of ''X'' such that ''S'' is contained in the union of the ''n'' translates of ''E'' by the points ''a''<sub>''i''</sub>.
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-A [[topological group]] ''X'' is ''left''-totally bounded if and only if it satisfies the definition for topological abelian groups above, using ''left'' translates. That is, use ''a''<sub>''i''</sub>''E'' in place of ''E'' + ''a''<sub>''i''</sub>. Alternatively, ''X'' is ''right''-totally bounded if and only if it satisfies the definition for topological abelian groups above, using ''right'' translates. That is, use ''Ea''<sub>''i''</sub> in place of ''E'' + ''a''<sub>''i''</sub>. (In other words, a "size" here is unambiguously a neighbourhood of the identity element, but there are two notions of ''whether'' a set is of a given size: a left notion based on left translation and a right notion based on right translation.)
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-Generalising the above definitions, a subset ''S'' of a [[uniform space]] ''X'' is totally bounded if and only if, given any [[entourage (topology)|entourage]] ''E'' in ''X'', there exists a finite cover of ''S'' by subsets of ''X'' each of whose [[Cartesian square]]s is a subset of ''E''. (In other words, a "size" here is an entourage, and a subset is of size ''E'' if its Cartesian square is a subset of ''E''.) Equivalently, ''S'' is totally bounded if and only if, given any ''E'' as before, there exist subsets ''A''<sub>1</sub>, ''A''<sub>2</sub>, ..., ''A''<sub>''n''</sub> of ''X'' such that ''S'' is contained in the union of the ''A''<sub>''i''</sub> and, whenever the elements ''x'' and ''y'' of ''X'' both belong to the same set ''A''<sub>''i''</sub>, then (''x'',''y'') belongs to ''E'' (so that ''x'' and ''y'' are close as measured by ''E'').
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-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 completion is compact.
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-== Examples and nonexamples ==
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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 set|bounded]]. Archimedean property is used.
-* The [[unit ball]] in a [[Hilbert space]], or more generally in a [[Banach space]], is totally bounded if and only if the space has finite [[dimension (linear algebra)|dimension]].
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-* Every [[compact set]] is totally bounded, whenever the concept is defined.
-* Every totally bounded metric space is bounded. However not every bounded metric space is totally bounded.<ref name=Willard>Willard, p. 182</ref>
+ </ul>
-* A subset of a [[complete metric space]] is totally bounded if and only if it is [[relatively compact set|relatively compact]] (meaning that its [[closure (topology)|closure]] is compact).
-* 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 set]]s.
-* A metric space is [[separable space|separable]] if and only if it is [[homeomorphism|homeomorphic]] to a totally bounded metric space.<ref name=Willard/>
-* An infinite metric space with the [[discrete metric]] (the distance between any two [[distinct]] points is 1) is ''not'' totally bounded, even though it is bounded.
-== Relationships with compactness and completeness ==
-There is a nice relationship between total boundedness and [[compactness (topology)|compactness]]:
-Every compact metric space is totally bounded.
-A uniform space is compact [[if and only if]] it is both totally bounded and [[Cauchy complete]]. This can be seen as a generalisation of the [[Heine–Borel theorem]] from [[Euclidean space]]s to arbitrary spaces: we must replace [[bounded set|boundedness]] with total boundedness (and also replace [[closed set|closedness]] with completeness).
-There is a complementary relationship between total boundedness and the process of [[Cauchy completion]]: A uniform space is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the fact that, in Euclidean spaces, a set is bounded if and only if its closure is bounded.)
-Combining these theorems, a uniform space is totally bounded if and only if its completion is compact. This may be taken as an alternative definition of total boundedness. Alternatively, this may be taken as a definition of ''precompactness'', while still using a separate definition of total boundedness. Then it becomes a theorem that a space is totally bounded if and only if it is precompact. (Separating the definitions in this way is useful in the absence of the [[axiom of choice]]; see the next section.)
-== Use of the axiom of choice ==
-The properties of total boundedness mentioned above rely in part on the [[axiom of choice]]. In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundedness in elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true (that is, the proof does not require choice) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded. But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
-==Notes==
-{{reflist}}
-== References ==
-*{{cite book | author=Willard, Stephen | title=General Topology | publisher=Dover Publications | year=2004 | isbn=0-486-43479-6}}
-*{{cite book | author=Sutherland, W.A. | title=Introduction to metric and topological spaces| publisher=Oxford University Press | year=1975 | isbn=0-19-853161-3 | zbl=0304.54002 }}
-{{DEFAULTSORT:Totally Bounded Space}}
-[[Category:Uniform spaces]]
-[[Category:Metric geometry]]
-[[Category:Compactness (mathematics)]]

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