Let X be a topological space and let x and y be points in X. We say that x and y can be separated if each lies in a neighborhood that does not contain the other point.
X is a T1 space if any two distinct points in X are separated.
X is an R0 space if any two topologically distinguishable points in X are separated.
A T1 space is also called an accessible space or a Tychonoff space, or a space with Fréchet topology and an R0 space is also called a symmetric space.
Properties
Let X be a topological space. Then the following conditions are equivalent:
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. Clearly, a space is T1 if and only if it's both R0 and T0. Note that a finite T1 space is necessarily discrete.
Examples
Sierpinski space is a simple example of a topology that is T0 but is not T1.
The cofinite topology on an infinite set is a simple example of a topology that is T1 but is not Hausdorff. 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 OA to be those subsets of X that contain all but a finite subsetA of X. Then given distinct integers x and y:
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 OA from the previous example, define a subbase of open sets Gx for any integerx to be Gx = O if x is an even number, and Gx = O if x is odd. Then the basis of the topology are given by finite intersections of the subbasis sets: given a finite set A, the open sets of X are
The Zariski topology on an algebraic variety is T1. To see this, note that a point with local coordinates is the zero set of the polynomials x1-c1,..., xn-cn. Thus, the point is closed. However, this example is well known as a space that is not Hausdorff. The Zariski topology is essentially an example of a cofinite topology.
The Zariski topology on a commutative 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. 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 of the topology is given by the open sets Oa of prime ideals that do not contain a in A. It is straightforward to verify that this indeed forms the basis: so Oa ∩ Ob = Oab and O0 = Ø and O1 = X. The closed sets of the Zariski topology are the sets of prime ideals that do 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.
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 are unique or unique up to topological indistinguishability. 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.
