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

_{1} space if any two distinct points in ''X'' are separated.
* ''X'' is called 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 entirely different meaning in _{1} space'' is preferred. There is also a notion of a Fréchet–Urysohn space as a type of _{1} space if and only if it is both an R_{0} space and a Kolmogorov (or T_{0}) space (i.e., a space in which distinct points are topologically distinguishable). A topological space is an R_{0} space if and only if its _{1} space.

_{1} space.
#$X$ is a T_{0} space and an R_{0} space.
#Points are closed in $X$; that is, for every point $x\; \backslash in\; X,$ the singleton set $\backslash $ is a _{0} space.
#Given any $x\; \backslash in\; X,$ the closure of $\backslash $ contains only the points that are topologically indistinguishable from $x.$
#The _{1}.
#For any $x,y\backslash in\; X,$ $x$ is in the closure of $\backslash $ if and only if $y$ is in the closure of $\backslash .$
#The specialization preorder on $X$ is _{0}. If the second arrow can be reversed the space is T_{0}. If the composite arrow can be reversed the space is T_{1}. A space is T_{1} if and only if it is both R_{0} and T_{0}.
A finite T_{1} space is necessarily discrete (since every set is closed).

_{0} but is not T_{1}, and hence also not R_{0}.
* 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 (T_{2}). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let $X$ be the set of _{1} by each of the definitions above. This space is not T_{2}, because the 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 but not 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 _{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 _{1}. To see this, note that the singleton containing a point with _{2}). The Zariski topology is essentially an example of a cofinite topology.
* The Zariski topology on a _{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 _{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 _{1} space, points are always closed.
* Every _{1}, since every point is a connected component and therefore closed.

_{1}", "R_{0}", and their synonyms can also be applied to such variations of topological spaces as _{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, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

and related branches of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Ttopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

in which, for every pair of distinct points, each has a neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural ...

not containing the other point. An Rtopologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...

points. The properties TDefinitions

Let ''X'' be atopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

and let ''x'' and ''y'' be points in ''X''. We say that ''x'' and ''y'' are if each lies in a neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

that does not contain the other point.
* ''X'' is called a Ttopologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...

points in ''X'' are separated.
A Tfunctional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

. For this reason, the term ''Tsequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...

. The term also has another meaning.)
A topological space is a TKolmogorov quotient
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing th ...

is a TProperties

If $X$ is a topological space then the following conditions are equivalent: #$X$ is a Tclosed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

of $X.$
#Every subset of $X$ is the intersection of all the open sets containing it.
#Every finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. ...

is closed.
#Every cofinite set of $X$ is open.
#For every $x\; \backslash in\; X,$ the fixed ultrafilter at $x$ converges only to $x.$
#For every subset $S$ of $X$ and every point $x\; \backslash in\; X,$ $x$ is a limit point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...

of $S$ if and only if every open neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of $x$ contains infinitely many points of $S.$
#Each map from the Sierpinski space to $X$ is trivial.
# The map from the Sierpinski space to the single point has the lifting property
In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given c ...

with respect to the map from $X$ to the single point.
If $X$ is a topological space then the following conditions are equivalent: (where $\backslash operatorname\backslash $ denotes the closure of $\backslash $)
#$X$ is an RKolmogorov quotient
In topology and related branches of mathematics, a topological space ''X'' is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of ''X'', at least one of them has a neighborhood not containing th ...

of $X$ is Tsymmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

(and therefore an equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...

).
#The sets $\backslash operatorname\backslash $ for $x\backslash in\; X$ form a partition of $X$ (that is, any two such sets are either identical or disjoint).
#If $F$ is a closed set and $x$ is a point not in $F$, then $F\backslash cap\backslash operatorname\backslash =\backslash emptyset.$
#Every neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of a point $x\backslash in\; X$ contains $\backslash operatorname\backslash .$
#Every open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...

is a union of closed sets.
#For every $x\; \backslash in\; X,$ the fixed ultrafilter at $x$ converges only to the points that are topologically indistinguishable from $x.$
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 RExamples

* Sierpinski space is a simple example of a topology that is Tinteger
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s, and define the open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...

s $O\_A$ to be those subsets of $X$ that contain all but a finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...

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 Tsubbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...

of open sets $G\_x$ for any integer $x$ to be $G\_x\; =\; O\_$ if $x$ is an even number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
4 ...

, and $G\_x\; =\; O\_$ if $x$ is odd. Then the basis of the topology are given by finite 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 Talgebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...

) is Tlocal coordinates
Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples:
* Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow ...

$\backslash left(c\_1,\; \backslash ldots,\; c\_n\backslash right)$ is the zero set of the polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...

s $x\_1\; -\; c\_1,\; \backslash ldots,\; x\_n\; -\; c\_n.$ Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (Tcommutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...

(that is, the prime spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...

) is Tprime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...

s that contain the point (and thus the topology is Tprime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...

s of $A.$ The 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 Ttotally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...

space is TGeneralisations to other kinds of spaces

The terms "Tuniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...

s, Cauchy space In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool deriv ...

s, and convergence spaces.
The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for TSee also

*Citations

Bibliography

* A.V. Arkhangel'skii, L.S. Pontryagin (Eds.) ''General Topology I'' (1990) Springer-Verlag . * * * 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). * {{DEFAULTSORT:T1 Space Properties of topological spaces Separation axioms