T1 space

In topology and related branches of mathematics, a T₁ space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. An R₀ space is one in which this holds for every pair of topologically distinguishable points. The properties T₁ and R₀ are examples of separation axioms.

Definitions

Let X be a topological space and let x and y be points in X. We say that x and y are if each lies in a neighbourhood that does not contain the other point.X is called a T₁ space if any two distinct points in X are separated.X is called an R₀ space if any two topologically distinguishable 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.
A topological space is a T₁ space if and only if it is both an R₀ space and a Kolmogorov (or T₀) space. A topological space is an R₀ space if and only if its Kolmogorov quotient is a T₁ space.

Properties

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

is a T₁ space.
is a T₀ space and an R₀ space.
Points are closed in ; that is, for every point the singleton set is a closed subset of
Every subset of is the intersection of all the open sets containing it.
Every finite set is closed.
Every cofinite set of is open.
For every the fixed ultrafilter at converges only to
For every subset of and every point is a limit point of if and only if every open neighbourhood of contains infinitely many points of
Each map from the Sierpiński space to is trivial.
The map from the Sierpiński 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 the closure of contains only the points that are topologically indistinguishable from
The Kolmogorov quotient of is T₁.
For any is in the closure of if and only if is in the closure of
The specialization preorder on is symmetric.
The sets for form a partition of .
If is a closed set and is a point not in, then
Every neighbourhood of a point contains
Every open set is a union of closed sets.
For every the fixed ultrafilter at converges only to the points that are topologically indistinguishable from

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 is both R₀ and T₀.
A finite T₁ space is necessarily discrete.
A space that is locally T₁, in the sense that each point has a T₁ neighbourhood, is also T₁. Similarly, a space that is locally R₀ is also R₀. In contrast, the corresponding statement does not hold for T₂ spaces. For example, the line with two origins is not a Hausdorff space but is locally Hausdorff.

Examples

Sierpiński space is a simple example of a topology that is T₀ but is not T₁, and hence also not R₀.
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. This follows since no two nonempty 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 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 Zariski topology on an algebraic variety is T₁. To see this, note that the singleton containing a point with local coordinates is the zero set of the polynomials 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 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 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 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.
Every totally disconnected space is T₁, since every point is a 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 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.