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| {{Onesource|date=June 2010}}
| | His name is Ezra. For years I've been living in Georgia and my parents live . She used to be unemployed and more so he can be a software beautiful. One of his favorite hobbies is watching movies and now he has time for taking on new things. I've been taking care of my website for ages now. You'll find the site here: http://www.cyberpublishers.net/design/cheap-jordans.php |
| {{DISPLAYTITLE:T<sub>1</sub> space}}
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| {{Separation axiom}}
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| In [[topology]] and related branches of [[mathematics]], a '''T<sub>1</sub> space''' is a [[topological space]] in which, for every pair of distinct points, each has an [[open neighborhood]] not containing the other. An '''R<sub>0</sub> space''' is one in which this holds for every pair of [[topologically distinguishable]] points. The properties T<sub>1</sub> and R<sub>0</sub> are examples of [[separation axiom]]s.
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| ==Definitions==
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| Let ''X'' be a [[topological space]] and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' can be ''[[separated sets|separated by open neighborhoods]]'' if each lies in an [[open set]] which does not contain the other point.
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| *''X'' is a '''T<sub>1</sub> space''' if any two [[distinct]] points in ''X'' can be separated by open neighborhoods.
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| *''X'' is a '''R<sub>0</sub> space''' if any two [[topologically distinguishable]] points in ''X'' can be separated by open neighborhoods.
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| A T<sub>1</sub> space is also called an '''accessible space''' or a '''Fréchet space''' and a R<sub>0</sub> space is also called a '''symmetric space'''. (The term ''Fréchet space'' also has an [[Fréchet space|entirely different meaning]] in [[functional analysis]]. For this reason, the term ''T<sub>1</sub> space'' is preferred. There is also a notion of a [[Fréchet-Urysohn space]] as a type of [[sequential space]]. The term ''symmetric space'' has [[Symmetric space|another meaning]].)
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| ==Properties==
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| Let ''X'' be a topological space. Then the following conditions are equivalent:
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| * ''X'' is a T<sub>1</sub> space.
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| * ''X'' is a [[Kolmogorov space|T<sub>0</sub> space]] and a R<sub>0</sub> space.
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| * Points are closed in ''X''; i.e. given any ''x'' in ''X'', the singleton set {''x''} is a [[closed set]].
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| * Every subset of ''X'' is the intersection of all the open sets containing it.
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| * Every [[finite set]] is closed.
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| * Every [[cofinite]] set of ''X'' is open.
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| * The [[fixed ultrafilter]] at ''x'' converges only to ''x''.
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| * For every point ''x'' in ''X'' and every subset ''S'' of ''X'', ''x'' is a [[limit point]] of ''S'' if and only if every open [[neighbourhood (topology)|neighbourhood]] of ''x'' contains infinitely many points of ''S''.
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| Let ''X'' be a topological space. Then the following conditions are equivalent:
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| * ''X'' is an R<sub>0</sub> space.
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| * Given any ''x'' in ''X'', the [[closure (topology)|closure]] of {''x''} contains only the points that ''x'' is topologically indistinguishable from.
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| * The [[specialization preorder]] on ''X'' is [[symmetric relation|symmetric]] (and therefore an [[equivalence relation]]).
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| * The fixed ultrafilter at ''x'' converges only to the points that ''x'' is topologically indistinguishable from.
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| *The [[Kolmogorov quotient]] of ''X'' (which identifies topologically indistinguishable points) is T<sub>1</sub>.
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| *Every [[open set]] is the union of [[closed sets]].
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| In any topological space we have, as properties of any two points, the following implications
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| :''separated'' ⇒ ''topologically distinguishable'' ⇒ ''distinct''
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| If the first arrow can be reversed the space is R<sub>0</sub>. If the second arrow can be reversed the space is [[T0 space|T<sub>0</sub>]]. If the composite arrow can be reversed the space is T<sub>1</sub>. Clearly, a space is T<sub>1</sub> if and only if it's both R<sub>0</sub> and T<sub>0</sub>.
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| Note that a finite T<sub>1</sub> space is necessarily [[discrete space|discrete]] (since every set is closed).
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| ==Examples==
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| * [[Sierpinski space]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>.
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| * The [[overlapping interval topology]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>.
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| * The [[cofinite topology]] on an [[infinite set]] is a simple example of a topology that is T<sub>1</sub> but is not [[Hausdorff space|Hausdorff]] (T<sub>2</sub>). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let ''X'' be the set of [[integer]]s, and define the [[open set]]s ''O''<sub>''A''</sub> to be those subsets of ''X'' which contain all but a [[finite set|finite]] subset ''A'' of ''X''. Then given distinct integers ''x'' and ''y'':
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| :* the open set ''O''<sub>{''x''}</sub> contains ''y'' but not ''x'', and the open set ''O''<sub>{''y''}</sub> contains ''x'' and not ''y'';
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| :* equivalently, every singleton set {''x''} is the complement of the open set ''O''<sub>{''x''}</sub>, so it is a closed set;
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| :so the resulting space is T<sub>1</sub> by each of the definitions above. This space is not T<sub>2</sub>, because the [[intersection (set theory)|intersection]] of any two open sets ''O''<sub>''A''</sub> and ''O''<sub>''B''</sub> is ''O''<sub>''A''∪''B''</sub>, which is never empty. Alternatively, the set of even integers is [[compact set|compact]] but not [[closed set|closed]], which would be impossible in a Hausdorff space.
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| * The above example can be modified slightly to create the [[double-pointed cofinite topology]], which is an example of an R<sub>0</sub> space that is neither T<sub>1</sub> nor R<sub>1</sub>. Let ''X'' be the set of integers again, and using the definition of ''O''<sub>''A''</sub> from the previous example, define a [[subbase]] of open sets ''G''<sub>''x''</sub> for any integer ''x'' to be ''G''<sub>''x''</sub> = ''O''<sub>{''x'', ''x''+1}</sub> if ''x'' is an [[even number]], and ''G''<sub>''x''</sub> = ''O''<sub>{''x''-1, ''x''}</sub> if ''x'' is odd. Then the [[basis (topology)|basis]] of the topology are given by finite [[intersection (set theory)|intersections]] of the subbasis sets: given a finite set ''A'', the open sets of ''X'' are
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| ::<math>U_A := \bigcap_{x \in A} G_x. </math>
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| :The resulting space is not T<sub>0</sub> (and hence not T<sub>1</sub>), because the points ''x'' and ''x'' + 1 (for ''x'' even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.
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| * The [[Zariski topology]] on an [[algebraic variety]] (over an [[algebraically closed field]]) is T<sub>1</sub>. To see this, note that a point with [[local coordinates]] (''c''<sub>1</sub>,...,''c''<sub>''n''</sub>) is the [[zero set]] of the [[polynomial]]s ''x''<sub>1</sub>-''c''<sub>1</sub>, ..., ''x''<sub>''n''</sub>-''c''<sub>''n''</sub>. Thus, the point is closed. However, this example is well known as a space that is not [[Hausdorff space|Hausdorff]] (T<sub>2</sub>). The Zariski topology is essentially an example of a cofinite topology.
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| * Every [[totally disconnected]] space is T<sub>1</sub>, since every point is a [[connected component (topology)|connected component]] and therefore closed.
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| ==Generalisations to other kinds of spaces==
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| The terms "T<sub>1</sub>", "R<sub>0</sub>", and their synonyms can also be applied to such variations of topological spaces as [[uniform space]]s, [[Cauchy space]]s, and [[convergence space]]s.
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| The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant [[net (topology)|net]]s) are unique (for T<sub>1</sub> spaces) or unique up to topological indistinguishability (for R<sub>0</sub> spaces).
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| As it turns out, uniform spaces, and more generally Cauchy spaces, are always R<sub>0</sub>, so the T<sub>1</sub> condition in these cases reduces to the T<sub>0</sub> condition.
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| But R<sub>0</sub> alone can be an interesting condition on other sorts of convergence spaces, such as [[pretopological space]]s.
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| == References ==
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| * {{Cite book| last=Willard| first=Stephen| year=1998| title=General Topology| edition=| volume= | series= |place=New York | publisher=Dover| pages=86–90| isbn = 0-486-43479-6}}.
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| * {{Cite book| last=Folland| first=Gerald| year=1999| title=Real analysis: modern techniques and their applications| edition=2nd| volume=| series=| place=| publisher=John Wiley & Sons, Inc| page=116| isbn = 0-471-31716-0}}.
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| {{DEFAULTSORT:T1 Space}}
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| [[Category:Topology]]
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| [[Category:Separation axioms]]
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| [[Category:Properties of topological spaces]]
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His name is Ezra. For years I've been living in Georgia and my parents live . She used to be unemployed and more so he can be a software beautiful. One of his favorite hobbies is watching movies and now he has time for taking on new things. I've been taking care of my website for ages now. You'll find the site here: http://www.cyberpublishers.net/design/cheap-jordans.php