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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 ...

topology
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 T1 space is a
topological 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 R0 space is one in which this holds for every pair of
topologically 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 T1 and R0 are examples of
separation 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 a
topological 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 T1 space if any two Distinct (mathematics), distinct points in ''X'' are separated. * ''X'' is an R0 space if any two
topologically 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 T1 space is also called an accessible space or a space with Fréchet topology and an R0 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 ''T1 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.)


Properties

If is a topological space then the following conditions are equivalent: # is a T1 space. # is a Kolmogorov space, T0 space and an R0 space. #Points are closed in ; i.e. given any x \in X, the singleton set \ 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 \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 R0 space. #Given any x \in X, the closure (topology), closure of \ contains only the points that are topologically indistinguishable from . #For any two points and in the space, is in the closure of \ if and only if is in the closure of \. #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 : \implies \implies If the first arrow can be reversed the space is R0. If the second arrow can be reversed the space is T0 space, T0. If the composite arrow can be reversed the space is T1. A space is T1 if and only if it's both R0 and T0. Note that a finite T1 space is necessarily discrete space, discrete (since every set is closed).


Examples

* Sierpinski space is a simple example of a topology that is T0 but is not T1. * The overlapping interval topology is a simple example of a topology that is T0 but is not T1. * Every weakly Hausdorff space is T1 but the converse is not true in general. * The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not Hausdorff space, Hausdorff (T2). 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 \ is the complement of the open set O_, so it is a closed set; :so the resulting space is T1 by each of the definitions above. This space is not T2, because the Intersection (set theory), intersection of any two open sets O_A and O_B is O_A \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 R0 space that is neither T1 nor R1. 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 := \bigcap_ G_x. :The resulting space is not T0 (and hence not T1), 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 T1. To see this, note that the singleton containing a point with local coordinates \left(c_1, \ldots, c_n\right) is the zero set of the polynomials x_1 - c_1, \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 (T2). 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 T0 but not, in general, T1.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 T0). 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 T1. 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 \in A. It is straightforward to verify that this indeed forms the basis: so O_a \cap O_b = O_ and O_0 = \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 T1 space, points are always closed. * Every totally disconnected space is T1, since every point is a Connected component (topology), connected component and therefore closed.


Generalisations to other kinds of spaces

The terms "T1", "R0", 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 T1 spaces) or unique up to topological indistinguishability (for R0 spaces). As it turns out, uniform spaces, and more generally Cauchy spaces, are always R0, so the T1 condition in these cases reduces to the T0 condition. But R0 alone can be an interesting condition on other sorts of convergence spaces, such as pretopological spaces.


See 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