Editing T1 space (section)

==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.