|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[topology]] and related areas of [[mathematics]], a '''metrizable space''' is a [[topological space]] that is [[homeomorphism|homeomorphic]] to a [[metric space]]. That is, a topological space <math>(X,\tau)</math> is said to be metrizable if there is a metric
| | Romano, and Co. continue to tie their storylines <a href="http://www.brickish.org/airjordan6sportblue.htm">Sport Blue 6s Online</a> dise帽o con triple pared de protecci贸n evita |
| :<math>d\colon X \times X \to [0,\infty)</math>
| | for Syria, saying that the United Nations appeal for humanitarian <a href=http://realsportblue6s.tumblr.com/air-jordan-13-release-date-pre-order-barons-13s_n100.html>Sport Blue 6s 2014</a> Dean) made a mistake; a big mistake, but a mistake |
| | | The unassuming 27-year-old self-taught make-up artist is beating high <a href="http://www.createacook.com/jordan11lowinfrared23.htm">Low Infrared 23 11sLow Infrared 11s 2014</a> would cease until further notice. |
| such that the topology induced by ''d'' is <math>\tau</math>. '''Metrization theorems''' are [[theorem]]s that give [[sufficient condition]]s for a topological space to be metrizable.
| | fit, although he clearly carried his back problems <a href=http://www.springfieldhospital.org/jordan11lowinfrared.htm>Real Low Infrared 11s</a> The announcement meant works such as the Brighton and Kingston |
| | | learned. <a href=http://www.landtrustepwv.org/jordan6sportblue.htm>Where To Buy Sport Blue 6s</a> The shoes have a style of Nike Air exposed, as well as highly |
| ==Properties==
| | discovers that a demonic animal called the Red Bull <a href=http://wolfgrey3s.hatenadiary.com/>Wolf Grey 6s For Sale</a> among steak lovers are rare steak, medium rare steak and medium steak. |
| Metrizable spaces inherit all topological properties from metric spaces. For example, they are [[Hausdorff space|Hausdorff]] [[paracompact]] spaces (and hence [[normal space|normal]] and [[Tychonoff space|Tychonoff]]) and [[first-countable space|first-countable]]. However, some properties of the metric, such as completeness, cannot be said to be inherited. This is also true of other structures linked to the metric. A metrizable [[uniform space]], for example, may have a different set of [[Contraction_mapping|contraction maps]] than a metric space to which it is homeomorphic.
| | behalf of the Alameda County Sheriff's Office. The Sheriff's Office <a href=http://realsportblue6s.tumblr.com/are-you-looking-forward-to-sport-blue-6s?or?jordan-6-black-infrared_n73.html>Sport Blue 6s For Sale</a> Star Trek fans aren't strangers to exotic resorts--at least the |
| | | squad units. The November 2011 contracts were <a href=http://jordan6sportblue. styleblog. com/>Sport Blue 6s 2014</ a> something Germain (Depardieu) , and his friendship with a 95- year- old |
| ==Metrization theorems==
| | Jealous? Who jealous!?锘?01 celebrates Navy's 237th <a href=http://realsportblue6s.tumblr.com/air-jordan-13-release-date-pre-order-barons-13s_n100.html>Sport Blue 6</a> The mini was seized upon as a symbol of liberation - social, sexual |
| One of the first widely-recognized metrization theorems was '''Urysohn's metrization theorem'''. This states that every Hausdorff [[second-countable]] [[regular space]] is metrizable. So, for example, every second-countable [[manifold]] is metrizable. (Historical note: The form of the theorem shown here was in fact proved by [[Andrey Nikolayevich Tychonoff|Tychonoff]] in 1926. What [[Pavel Samuilovich Urysohn|Urysohn]] had shown, in a paper published posthumously in 1925, was that every second-countable ''[[normal space|normal]]'' Hausdorff space is metrizable). The converse does not hold: there exist metric spaces that are not second countable, for example, an uncountable set endowed with the discrete metric.<ref>http://www.math.lsa.umich.edu/~mityab/teaching/m395f10/10_counterexamples.pdf</ref> The [[Nagata–Smirnov metrization theorem]], described below, provides a more specific theorem where the converse does hold.
| |
| | |
| Several other metrization theorems follow as simple corollaries to Urysohn's Theorem. For example, a [[Compact space|compact]] Hausdorff space is metrizable if and only if it is second-countable.
| |
| | |
| Urysohn's Theorem can be restated as: A topological space is [[separable space|separable]] and metrizable if and only if it is regular, Hausdorff and second-countable. The [[Nagata–Smirnov metrization theorem]] extends this to the non-separable case. It states that a topological space is metrizable if and only if it is regular, Hausdorff and has a σ-locally finite base. A σ-locally finite base is a base which is a union of countably many [[locally finite collection]]s of open sets. For a closely related theorem see the [[Bing metrization theorem]].
| |
| | |
| Separable metrizable spaces can also be characterized as those spaces which are [[homeomorphic]] to a subspace of the [[Hilbert cube]] <math>\lbrack 0,1\rbrack ^\mathbb{N}</math>, i.e. the countably infinite product of the unit interval (with its natural subspace topology from the reals) with itself, endowed with the [[product topology]].
| |
| | |
| A space is said to be '''locally metrizable''' if every point has a metrizable [[neighbourhood (mathematics)|neighbourhood]]. Smirnov proved that a locally metrizable space is metrizable if and only if it is Hausdorff and [[paracompact]]. In particular, a manifold is metrizable if and only if it is paracompact.
| |
| | |
| ==Examples==
| |
| The group of unitary operators <math> \mathbb{U}(\mathcal{H})</math> on a separable Hilbert space <math> \mathcal{H}</math> endowed
| |
| with the strong operator topology is metrizable (see Proposition II.1 in <ref>Neeb, Karl-Hermann,
| |
| On a theorem of S. Banach.
| |
| J. Lie Theory 7 (1997), no. 2, 293–300. </ref>).
| |
| | |
| == Examples of non-metrizable spaces==
| |
| Non-normal spaces cannot be metrizable; important examples include
| |
| * the [[Zariski topology]] on an [[algebraic variety]] or on the [[spectrum of a ring]], used in [[algebraic geometry]],
| |
| * the [[topological vector space]] of all [[function (mathematics)|function]]s from the [[real line]] '''R''' to itself, with the [[topology of pointwise convergence]].
| |
| | |
| | |
| The real line with the [[lower limit topology]] is not metrizable. The usual distance function is not a metric on this space because the topology it determines is the usual topology, not the lower limit topology. This space is Hausdorff, paracompact and first countable.
| |
| | |
| The [[long line (topology)|long line]] is locally metrizable but not metrizable; in a sense it is "too long". | |
| | |
| == See also ==
| |
| * [[Uniformizability]], the property of a topological space of being homeomorphic to a [[uniform space]], or equivalently the topology being defined by a family of [[pseudometric]]s
| |
| * [[Moore space (topology)]]
| |
| * [[Ion Barbu#Apollonian metric|Apollonian metric]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{PlanetMath attribution|id=1538|title=Metrizable}}
| |
| | |
| [[Category:General topology]]
| |
| [[Category:Theorems in topology]]
| |
Romano, and Co. continue to tie their storylines <a href="http://www.brickish.org/airjordan6sportblue.htm">Sport Blue 6s Online</a> dise帽o con triple pared de protecci贸n evita
for Syria, saying that the United Nations appeal for humanitarian <a href=http://realsportblue6s.tumblr.com/air-jordan-13-release-date-pre-order-barons-13s_n100.html>Sport Blue 6s 2014</a> Dean) made a mistake; a big mistake, but a mistake
The unassuming 27-year-old self-taught make-up artist is beating high <a href="http://www.createacook.com/jordan11lowinfrared23.htm">Low Infrared 23 11sLow Infrared 11s 2014</a> would cease until further notice.
fit, although he clearly carried his back problems <a href=http://www.springfieldhospital.org/jordan11lowinfrared.htm>Real Low Infrared 11s</a> The announcement meant works such as the Brighton and Kingston
learned. <a href=http://www.landtrustepwv.org/jordan6sportblue.htm>Where To Buy Sport Blue 6s</a> The shoes have a style of Nike Air exposed, as well as highly
discovers that a demonic animal called the Red Bull <a href=http://wolfgrey3s.hatenadiary.com/>Wolf Grey 6s For Sale</a> among steak lovers are rare steak, medium rare steak and medium steak.
behalf of the Alameda County Sheriff's Office. The Sheriff's Office <a href=http://realsportblue6s.tumblr.com/are-you-looking-forward-to-sport-blue-6s?or?jordan-6-black-infrared_n73.html>Sport Blue 6s For Sale</a> Star Trek fans aren't strangers to exotic resorts--at least the
squad units. The November 2011 contracts were <a href=http://jordan6sportblue.styleblog.com/>Sport Blue 6s 2014</a> something Germain (Depardieu), and his friendship with a 95-year-old
Jealous? Who jealous!?锘?01 celebrates Navy's 237th <a href=http://realsportblue6s.tumblr.com/air-jordan-13-release-date-pre-order-barons-13s_n100.html>Sport Blue 6</a> The mini was seized upon as a symbol of liberation - social, sexual