Editing T1 space

{{Short description|Topological space in which all singleton sets are closed}}
{{DISPLAYTITLE:T<sub>1</sub> space}}
{{Separation axioms}}

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 a [[neighbourhood (mathematics)|neighborhood]] not containing the other point.<ref>Arkhangel'skii (1990). ''See section 2.6.''</ref>  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.

==Definitions==
Let ''X'' be a [[topological space]] and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are {{em|[[Separated sets|separated]]}} if each lies in a [[neighbourhood (mathematics)|neighbourhood]] that does not contain the other point.
* ''X'' is called a '''T<sub>1</sub> space''' if any two distinct points in ''X'' are separated.
* ''X'' is called an '''R<sub>0</sub> space''' if any two [[topologically distinguishable]] points in ''X'' are separated.

A T<sub>1</sub> space is also called an '''accessible space''' or a space with '''Fréchet topology''' and an R<sub>0</sub> space is also called a '''symmetric space'''. (The term {{em|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 {{em|symmetric space}} also has [[Symmetric space|another meaning]].)

A topological space is a T<sub>1</sub> space if and only if it is both an R<sub>0</sub> space and a [[Kolmogorov space|Kolmogorov (or T<sub>0</sub>) space]] (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R<sub>0</sub> space if and only if its [[Kolmogorov quotient]] is a T<sub>1</sub> space.

==Properties==

If <math>X</math> is a topological space then the following conditions are equivalent:

#<math>X</math> is a T<sub>1</sub> space.
#<math>X</math> is a [[Kolmogorov space|T<sub>0</sub> space]] and an R<sub>0</sub> space.
#Points are closed in <math>X</math>; that is, for every point <math>x \in X,</math> the singleton set <math>\{x\}</math> is a [[Closed set|closed subset]] of <math>X.</math>
#Every subset of <math>X</math> is the intersection of all the open sets containing it.
#Every [[finite set]] is closed.<ref>Archangel'skii (1990) ''See proposition 13, section 2.6.''</ref>
#Every [[cofinite]] set of <math>X</math> is open.
#For every <math>x \in X,</math> the [[fixed ultrafilter]] at <math>x</math> [[Convergent filter|converges]] only to <math>x.</math>
#For every subset <math>S</math> of <math>X</math> and every point <math>x \in X,</math> <math>x</math> is a [[Limit point of a set|limit point]] of <math>S</math> if and only if every open [[Neighbourhood (topology)|neighbourhood]] of <math>x</math> contains infinitely many points of <math>S.</math>
#Each map from the [[Sierpiński space]] to <math>X</math> is trivial.
# The map from the [[Sierpiński space]] to the single point has the [[lifting property]] with respect to the map from <math>X</math> to the single point. 

If <math>X</math> is a topological space then the following conditions are equivalent:{{sfn|Schechter|1996|loc=16.6, p. 438}} (where <math>\operatorname{cl}\{x\}</math> denotes the closure of <math>\{x\}</math>)

#<math>X</math> is an R<sub>0</sub> space.
#Given any <math>x \in X,</math> the [[Closure (topology)|closure]] of <math>\{x\}</math> contains only the points that are topologically indistinguishable from <math>x.</math>
#The [[Kolmogorov quotient]] of <math>X</math> is T<sub>1</sub>.
#For any <math>x,y\in X,</math> <math>x</math> is in the closure of <math>\{y\}</math> if and only if <math>y</math> is in the closure of <math>\{x\}.</math>
#The [[specialization preorder]] on <math>X</math> is [[Symmetric relation|symmetric]] (and therefore an [[equivalence relation]]).
#The sets <math>\operatorname{cl}\{x\}</math> for <math>x\in X</math> form a [[partition (set theory)|partition]] of <math>X</math> (that is, any two such sets are either identical or disjoint).
#If <math>F</math> is a closed set and <math>x</math> is a point not in <math>F</math>, then <math>F\cap\operatorname{cl}\{x\}=\emptyset.</math>
#Every [[neighbourhood (mathematics)|neighbourhood]] of a point <math>x\in X</math> contains <math>\operatorname{cl}\{x\}.</math>
#Every [[open set]] is a union of [[closed set]]s.
#For every <math>x \in X,</math> the fixed ultrafilter at <math>x</math> converges only to the points that are topologically indistinguishable from <math>x.</math>

In any topological space we have, as properties of any two points, the following implications

:<math>\text{separated}\implies\text{topologically distinguishable}\implies\text{distinct.}</math>

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>. A space is T<sub>1</sub> if and only if it is both R<sub>0</sub> and T<sub>0</sub>.

A finite T<sub>1</sub> space is necessarily [[discrete space|discrete]] (since every set is closed).

A space that is locally T<sub>1</sub>, in the sense that each point has a T<sub>1</sub> neighbourhood (when given the subspace topology), is also T<sub>1</sub>.<ref>{{cite web |title=Locally Euclidean space implies T1 space |url=https://math.stackexchange.com/questions/3142975 |website=Mathematics Stack Exchange}}</ref>  Similarly, a space that is locally R<sub>0</sub> is also R<sub>0</sub>.  In contrast, the corresponding statement does not hold for T<sub>2</sub> spaces.  For example, the [[line with two origins]] is not a [[Hausdorff space]] but is locally Hausdorff.

== Examples ==
* [[Sierpiński space]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>, and hence also not R<sub>0</sub>.
* The [[overlapping interval topology]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>.
* Every [[weakly Hausdorff space]] is T<sub>1</sub> but the converse is not true in general. 
* 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 nonempty open sets of the cofinite topology are disjoint.  Specifically, let <math>X</math> be the set of [[integer]]s, and define the [[open set]]s <math>O_A</math> to be those subsets of <math>X</math> that contain all but a [[Finite set|finite]] subset <math>A</math> of <math>X.</math> Then given distinct integers <math>x</math> and <math>y</math>:
:* the open set <math>O_{\{ x \}}</math> contains <math>y</math> but not <math>x,</math> and the open set <math>O_{\{ y \}}</math> contains <math>x</math> and not <math>y</math>;
:* equivalently, every singleton set <math>\{ x \}</math> is the complement of the open set <math>O_{\{ x \}},</math> so it is a closed set;
: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 <math>O_A</math> and <math>O_B</math> is <math>O_A \cap O_B = O_{A \cup B},</math> 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.

* 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 <math>X</math> be the set of integers again, and using the definition of <math>O_A</math> from the previous example, define a [[subbase]] of open sets <math>G_x</math> for any integer <math>x</math> to be <math>G_x = O_{\{ x, x+1 \}}</math> if <math>x</math> is an [[even number]], and <math>G_x = O_{\{ x-1, x \}}</math> if <math>x</math> is odd.  Then the [[Basis (topology)|basis]] of the topology are given by finite [[Intersection (set theory)|intersections]] of the subbasic sets: given a finite set <math>A,</math>the open sets of <math>X</math> are

::<math>U_A := \bigcap_{x \in A} G_x. </math>

:The resulting space is not T<sub>0</sub> (and hence not T<sub>1</sub>), because the points <math>x</math> and <math>x + 1</math> (for <math>x</math> even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.

* The [[Zariski topology]] on an [[algebraic variety]] (over an [[algebraically closed field]]) is T<sub>1</sub>.  To see this, note that the singleton containing a point with [[local coordinates]] <math>\left(c_1, \ldots, c_n\right)</math> is the [[zero set]] of the [[polynomial]]s <math>x_1 - c_1, \ldots, x_n - c_n.</math>  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.
* The Zariski topology on a [[commutative ring]] (that is, the prime [[spectrum of a ring]]) is T<sub>0</sub> but not, in general, T<sub>1</sub>.<ref>Arkhangel'skii (1990). ''See example 21, section 2.6.''</ref>  To see this, note that the closure of a one-point set is the set of all [[prime ideal]]s that contain the point (and thus the topology is T<sub>0</sub>). However, this closure is a [[maximal ideal]], and the only closed points are the maximal ideals, and are thus not contained in any of the open sets of the topology, and thus the space does not satisfy axiom T<sub>1</sub>.  To be clear about this example: the Zariski topology for a commutative ring <math>A</math> is given as follows: the topological space is the set <math>X</math> of all [[prime ideal]]s of <math>A.</math> The [[base (topology)|base of the topology]] is given by the open sets <math>O_a</math> of prime ideals that do {{em|not}} contain <math>a \in A.</math> It is straightforward to verify that this indeed forms the basis: so <math>O_a \cap O_b = O_{ab}</math> and <math>O_0 = \varnothing</math> and <math>O_1 = X.</math> The closed sets of the Zariski topology are the sets of prime ideals that {{em|do}} contain <math>a.</math> Notice how this example differs subtly from the cofinite topology example, above: the points in the topology are not closed, in general, whereas in a  T<sub>1</sub> space, points are always closed.
* Every [[totally disconnected]] space is T<sub>1</sub>, since every point is a [[Connected component (topology)|connected component]] and therefore closed.

==Generalisations to other kinds of spaces==
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.
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).

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.
But R<sub>0</sub> alone can be an interesting condition on other sorts of convergence spaces, such as [[pretopological space]]s.

== See also ==

* {{annotated link|Topological property}}

== Citations ==
{{reflist}}

==Bibliography==

* A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) ''General Topology I'' (1990) Springer-Verlag {{isbn|3-540-18178-4}}.
* {{Cite book| last=Folland| first=Gerald| year=1999| title=Real analysis: modern techniques and their applications| url=https://archive.org/details/realanalysismode00foll_670| url-access=limited| edition=2nd| publisher=John Wiley & Sons, Inc| page=[https://archive.org/details/realanalysismode00foll_670/page/n128 116]| isbn = 0-471-31716-0}}
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. {{ISBN|0-486-68735-X}} (Dover edition).
* {{Cite book| last=Willard| first=Stephen| year=1998| title=General Topology|place=New York | publisher=Dover| pages=86–90| isbn = 0-486-43479-6}}

{{Topology}}

{{DEFAULTSORT:T1 Space}}

[[Category:Properties of topological spaces]]
[[Category:Separation axioms]]