In any topological space we have, as properties of any two points, the following implications
If the first arrow can be reversed the space is R0. If the second arrow can be reversed the space is 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 (since every set is closed).
The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not Hausdorff (T2). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let be the set of integers, and define the open sets to be those subsets of that contain all but a finite subset of Then given distinct integers and :
the open set contains but not and the open set contains and not ;
equivalently, every singleton set is the complement of the open set 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 of any two open sets and is 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 R0 space that is neither T1 nor R1. Let be the set of integers again, and using the definition of from the previous example, define a subbase of open sets for any integer to be if is an even number, and if is odd. Then the basis of the topology are given by finite intersections of the subbasic sets: given a finite set the open sets of are
The resulting space is not T0 (and hence not T1), because the points and (for even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.
The Zariski topology on a commutative ring (that is, the prime spectrum of a ring) is T0 but not, in general, T1. 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 is given as follows: the topological space is the set of all prime ideals of The base of the topology is given by the open sets of prime ideals that do not contain It is straightforward to verify that this indeed forms the basis: so and and The closed sets of the Zariski topology are the sets of prime ideals that do contain 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.
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 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.