In _{1} space is a _{0} space is one in which this holds for every pair of _{1} and R_{0} are examples of

_{1} space if any two Distinct (mathematics), distinct points in ''X'' are separated.
* ''X'' is an R_{0} space if any two _{1} space is also called an accessible space or a space with Fréchet topology and an R_{0} space is also called a symmetric space. (The term also has an Fréchet space, entirely different meaning in functional analysis. For this reason, the term ''T_{1} space'' is preferred. There is also a notion of a Fréchet–Urysohn space as a type of sequential space. The term has Symmetric space, another meaning.)

_{1} space.
# is a Kolmogorov space, T_{0} space and an R_{0} space.
#Points are closed in ; i.e. given any $x\; \backslash in\; X,$ the singleton set $\backslash $ is a closed set.
#Every subset of is the intersection of all the open sets containing it.
#Every finite set is closed.
#Every cofinite set of is open.
#The fixed ultrafilter at converges only to .
#For every subset of and every point $x\; \backslash in\; X,$ is a limit point of if and only if every open neighbourhood (topology), neighbourhood of contains infinitely many points of .
#Each map from the Sierpinski space to is trivial.
# The map from the Sierpinski space to the single point has the lifting property with respect to the map from to the single point.
If is a topological space then the following conditions are equivalent:
# is an R_{0} space.
#Given any $x\; \backslash in\; X,$ the closure (topology), closure of $\backslash $ contains only the points that are topologically indistinguishable from .
#For any two points and in the space, is in the closure of $\backslash $ if and only if is in the closure of $\backslash .$
#The specialization preorder on is symmetric relation, symmetric (and therefore an equivalence relation).
#The fixed ultrafilter at converges only to the points that are topologically indistinguishable from .
#Every open set is the union of closed sets.
In any topological space we have, as properties of any two points, the following implications
: $\backslash implies$ $\backslash implies$
If the first arrow can be reversed the space is R_{0}. If the second arrow can be reversed the space is T0 space, T_{0}. If the composite arrow can be reversed the space is T_{1}. A space is T_{1} if and only if it's both R_{0} and T_{0}.
Note that a finite T_{1} space is necessarily discrete space, discrete (since every set is closed).

_{0} but is not T_{1}.
* The overlapping interval topology is a simple example of a topology that is T_{0} but is not T_{1}.
* Every weakly Hausdorff space is T_{1} 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_{1} but is not Hausdorff space, Hausdorff (T_{2}). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let $X$ be the set of integers, and define the open sets $O\_A$ to be those subsets of $X$ that contain all but a Finite set, finite subset $A$ of $X.$ Then given distinct integers $x$ and $y$:
:* the open set $O\_$ contains $y$ but not $x,$ and the open set $O\_$ contains $x$ and not $y$;
:* equivalently, every singleton set $\backslash $ is the complement of the open set $O\_,$ so it is a closed set;
:so the resulting space is T_{1} by each of the definitions above. This space is not T_{2}, because the Intersection (set theory), intersection of any two open sets $O\_A$ and $O\_B$ is $O\_A\; \backslash cap\; O\_B\; =\; O\_,$ 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_{0} space that is neither T_{1} nor R_{1}. Let $X$ be the set of integers again, and using the definition of $O\_A$ from the previous example, define a subbase of open sets $G\_x$ for any integer $x$ to be $G\_x\; =\; O\_$ if $x$ is an even number, and $G\_x\; =\; O\_$ if $x$ 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 $A,$the open sets of $X$ are
::$U\_A\; :=\; \backslash bigcap\_\; G\_x.$
:The resulting space is not T_{0} (and hence not T_{1}), because the points $x$ and $x\; +\; 1$ (for $x$ 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_{1}. To see this, note that the singleton containing a point with local coordinates $\backslash left(c\_1,\; \backslash ldots,\; c\_n\backslash right)$ is the zero set of the polynomials $x\_1\; -\; c\_1,\; \backslash ldots,\; x\_n\; -\; c\_n.$ Thus, the point is closed. However, this example is well known as a space that is not Hausdorff space, Hausdorff (T_{2}). 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_{0} but not, in general, T_{1}.Arkhangel'skii (1990). ''See example 21, section 2.6.'' To see this, note that the closure of a one-point set is the set of all prime ideals that contain the point (and thus the topology is T_{0}). 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_{1}. To be clear about this example: the Zariski topology for a commutative ring $A$ is given as follows: the topological space is the set $X$ of all prime ideals of $A.$ The base (topology), base of the topology is given by the open sets $O\_a$ of prime ideals that do contain $a\; \backslash in\; A.$ It is straightforward to verify that this indeed forms the basis: so $O\_a\; \backslash cap\; O\_b\; =\; O\_$ and $O\_0\; =\; \backslash varnothing$ and $O\_1\; =\; X.$ The closed sets of the Zariski topology are the sets of prime ideals that contain $a.$ 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_{1} space, points are always closed.
* Every totally disconnected space is T_{1}, since every point is a Connected component (topology), connected component and therefore closed.

_{1}", "R_{0}", and their synonyms can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, and convergence spaces.
The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant net (topology), nets) are unique (for T_{1} spaces) or unique up to topological indistinguishability (for R_{0} spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always R_{0}, so the T_{1} condition in these cases reduces to the T_{0} condition.
But R_{0} alone can be an interesting condition on other sorts of convergence spaces, such as pretopological spaces.

topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and related branches of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a Ttopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

in which, for every pair of distinct points, each has a neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

not containing the other point. An Rtopologically distinguishable
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...

points. The properties Tseparation axiom
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

s.
Definitions

Let ''X'' be atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' can be if each lies in a neighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

that does not contain the other point.
* ''X'' is a Ttopologically distinguishable
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...

points in ''X'' are separated.
A TProperties

If is a topological space then the following conditions are equivalent: # is a TExamples

* Sierpinski space is a simple example of a topology that is TGeneralisations to other kinds of spaces

The terms "TSee also

*Citations

Bibliography

* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition). * * * A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) ''General Topology I'' (1990) Springer-Verlag . {{DEFAULTSORT:T1 Space Separation axioms Properties of topological spaces