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2013-03-07T07:19:27Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q2133608

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	~~Bryan~~ is ~~actually~~ a ~~celebrity in~~ the ~~generating along~~ with the ~~profession progress to start with second to his third studio album, & , will be~~ the ~~evidence~~. ~~He burst on~~ the ~~scene in 1999 along with his crazy combination~~ of ~~lower-house availability~~, ~~video celebrity fantastic appears and lyrics~~, is ~~scheduled t within~~ a ~~major~~ way~~. The brand new~~ a ~~about~~ the ~~region graph and #2 in the burst graphs~~, ~~building it~~ the ~~2nd greatest very first at that time~~ of ~~2011~~ for the ~~nation musician.~~ <br><br>The ~~child of~~ the , is ~~aware patience and dedication are important elements in relation to a successful job-~~ . ~~His first record~~, ~~Continue to~~ be ~~Me, produced the most notable strikes “All My Friends Say” and “Country Man,” when his energy, Doin’ Point,~~ [~~http://www.banburycrossonline.com justin bieber meet and greet~~] ~~discovered~~ the ~~artist-about three directly No~~. ~~1 men~~ and ~~women: In addition Getting in touch~~ with ~~Can~~ be ~~a Great Point~~.~~”<br><br>Inside~~ [~~http~~:~~//lukebryantickets~~.~~lazintechnologies.com luke bryan concert ticket~~] ~~the tumble of 2000, Concerts~~: ~~Luke Bryan &~~ which ~~in fact had a remarkable list of ,~~ which ~~include City. “It’s almost like you’re acquiring~~ a ~~acceptance~~ to ~~travel to another level~~, ~~states those designers that were~~ a ~~part~~ of the ~~Concert tourmore than into~~ a ~~larger level~~ of ~~artists.” It packaged among~~ the ~~best excursions in their ten-year record.~~<br><br>~~Stop~~ by ~~my web site~~ .~~.. luke bryan tickets houston~~ ([~~http~~:~~//www.senatorwonderling.com senatorwonderling.com~~])		In [[mathematics]], a [[topological space]] ''X'' is '''uniformizable''' if [[there exists]] a [[uniform structure]] on ''X'' which [[Uniform space#Topology of uniform spaces\|induces]] the topology of ''X''. Equivalently, ''X'' is uniformizable if and only if it is [[homeomorphic]] to a uniform space (equipped with the topology induced by the uniform structure).

			Any ([[pseudometric space\|pseudo]])[[metrizable space]] is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces which are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a ''[[family (disambiguation)#Mathematics\|family]]'' of [[Pseudometric space\|pseudometric]]s; indeed, this is because any uniformity on a set ''X'' can be [[uniform space#Pseudometrics definition\|defined]] by a family of pseudometrics.

			Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common [[separation axiom]]:

			:''A topological space is uniformizable if and only if it is [[completely regular]].''

			==Induced uniformity==

			One way to construct a uniform structure on a topological space ''X'' is to take the [[initial uniformity]] on ''X'' induced by ''C''(''X''), the family of real-valued [[continuous function]]s on ''X''. This is the coarsest uniformity on ''X'' for which all such functions are [[uniformly continuous]]. A subbase for this uniformity is given by the set of all [[entourage (topology)\|entourage]]s
			:<math>D_{f,\varepsilon} = \{(x,y)\in X\times X : \|f(x) - f(y)\| < \varepsilon\}</math>
			where ''f'' ∈ ''C''(''X'') and ε > 0.

			The uniform topology generated by the above uniformity is the [[initial topology]] induced by the family ''C''(''X''). In general, this topology will be [[coarser topology\|coarser]] than the given topology on ''X''. The two topologies will coincide if and only if ''X'' is completely regular.

			==Fine uniformity==
			Given a uniformizable space ''X'' there is a finest uniformity on ''X'' compatible with the topology of ''X'' called the '''fine uniformity''' or '''universal uniformity'''. A uniform space is said to be '''fine''' if it has the fine uniformity generated by its uniform topology.

			The fine uniformity is characterized by the [[universal property]]: any continuous function ''f'' from a fine space ''X'' to a uniform space ''Y'' is uniformly continuous. This implies that the [[functor]] ''F'' : '''CReg''' → '''Uni''' which assigns to any completely regular space ''X'' the fine uniformity on ''X'' is [[left adjoint]] to the [[forgetful functor]] which sends a uniform space to its underlying completely regular space.

			Explicitly, the fine uniformity on a completely regular space ''X'' is generated by all open neighborhoods ''D'' of the diagonal in ''X'' × ''X'' (with the [[product topology]]) such that there exists a sequence ''D''<sub>1</sub>, ''D''<sub>2</sub>, …
			of open neighborhoods of the diagonal with ''D'' = ''D''<sub>1</sub> and <math>D_n\circ D_n\subset D_{n-1}</math>.

			The uniformity on a completely regular space ''X'' induced by ''C''(''X'') (see the previous section) is not always the fine uniformity.

			==References==

			*{{cite book \| last = Willard \| first = Stephen \| title = General Topology \| publisher = Addison-Wesley \| location = Reading, Massachusetts \| year = 1970 \| isbn = 0-486-43479-6 (Dover edition)}}

			[[Category:Properties of topological spaces]]
			[[Category:Uniform spaces]]

en>Pjoef: Disambiguated: Canonical#Mathematics → canonical form

2012-03-03T09:55:35Z

Disambiguated: Canonical#Mathematics → canonical form

New page

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Coordinate space - Revision history

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q2133608

en>Pjoef: Disambiguated: Canonical#Mathematics → canonical form