Editing T1 space (section)

== Examples ==
* [[Sierpiński space]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>, and hence also not R<sub>0</sub>.
* The [[overlapping interval topology]] is a simple example of a topology that is T<sub>0</sub> but is not T<sub>1</sub>.
* Every [[weakly Hausdorff space]] is T<sub>1</sub> 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<sub>1</sub> but is not [[Hausdorff space|Hausdorff]] (T<sub>2</sub>). This follows since no two nonempty open sets of the cofinite topology are disjoint.  Specifically, let <math>X</math> be the set of [[integer]]s, and define the [[open set]]s <math>O_A</math> to be those subsets of <math>X</math> that contain all but a [[Finite set|finite]] subset <math>A</math> of <math>X.</math> Then given distinct integers <math>x</math> and <math>y</math>:
:* the open set <math>O_{\{ x \}}</math> contains <math>y</math> but not <math>x,</math> and the open set <math>O_{\{ y \}}</math> contains <math>x</math> and not <math>y</math>;
:* equivalently, every singleton set <math>\{ x \}</math> is the complement of the open set <math>O_{\{ x \}},</math> so it is a closed set;
:so the resulting space is T<sub>1</sub> by each of the definitions above.  This space is not T<sub>2</sub>, because the [[Intersection (set theory)|intersection]] of any two open sets <math>O_A</math> and <math>O_B</math> is <math>O_A \cap O_B = O_{A \cup B},</math> 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 R<sub>0</sub> space that is neither T<sub>1</sub> nor R<sub>1</sub>.  Let <math>X</math> be the set of integers again, and using the definition of <math>O_A</math> from the previous example, define a [[subbase]] of open sets <math>G_x</math> for any integer <math>x</math> to be <math>G_x = O_{\{ x, x+1 \}}</math> if <math>x</math> is an [[even number]], and <math>G_x = O_{\{ x-1, x \}}</math> if <math>x</math> 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 <math>A,</math>the open sets of <math>X</math> are

::<math>U_A := \bigcap_{x \in A} G_x. </math>

:The resulting space is not T<sub>0</sub> (and hence not T<sub>1</sub>), because the points <math>x</math> and <math>x + 1</math> (for <math>x</math> 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 T<sub>1</sub>.  To see this, note that the singleton containing a point with [[local coordinates]] <math>\left(c_1, \ldots, c_n\right)</math> is the [[zero set]] of the [[polynomial]]s <math>x_1 - c_1, \ldots, x_n - c_n.</math>  Thus, the point is closed.  However, this example is well known as a space that is not [[Hausdorff space|Hausdorff]] (T<sub>2</sub>).  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 T<sub>0</sub> but not, in general, T<sub>1</sub>.<ref>Arkhangel'skii (1990). ''See example 21, section 2.6.''</ref>  To see this, note that the closure of a one-point set is the set of all [[prime ideal]]s that contain the point (and thus the topology is T<sub>0</sub>). 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<sub>1</sub>.  To be clear about this example: the Zariski topology for a commutative ring <math>A</math> is given as follows: the topological space is the set <math>X</math> of all [[prime ideal]]s of <math>A.</math> The [[base (topology)|base of the topology]] is given by the open sets <math>O_a</math> of prime ideals that do {{em|not}} contain <math>a \in A.</math> It is straightforward to verify that this indeed forms the basis: so <math>O_a \cap O_b = O_{ab}</math> and <math>O_0 = \varnothing</math> and <math>O_1 = X.</math> The closed sets of the Zariski topology are the sets of prime ideals that {{em|do}} contain <math>a.</math> 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  T<sub>1</sub> space, points are always closed.
* Every [[totally disconnected]] space is T<sub>1</sub>, since every point is a [[Connected component (topology)|connected component]] and therefore closed.