Editing Kolmogorov space (section)

===Spaces that are T<sub>0</sub> but not T<sub>1</sub>===

*The [[Zariski topology]] on Spec(''R''), the [[prime spectrum]] of a [[commutative ring]] ''R'', is always T<sub>0</sub> but generally not T<sub>1</sub>. The non-closed points correspond to [[prime ideal]]s which are not [[maximal ideal|maximal]]. They are important to the understanding of [[scheme (mathematics)|scheme]]s.
*The [[particular point topology]] on any set with at least two elements is T<sub>0</sub> but not T<sub>1</sub> since the particular point is not closed (its closure is the whole space). An important special case is the [[Sierpiński space]] which is the particular point topology on the set {0,1}.
*The [[excluded point topology]] on any set with at least two elements is T<sub>0</sub> but not T<sub>1</sub>. The only closed point is the excluded point.
*The [[Alexandrov topology]] on a [[partially ordered set]] is T<sub>0</sub> but will not be T<sub>1</sub> unless the order is discrete (agrees with equality). Every finite T<sub>0</sub> space is of this type. This also includes the particular point and excluded point topologies as special cases.
*The [[right order topology]] on a [[totally ordered set]] is a related example.
*The [[overlapping interval topology]] is similar to the particular point topology since every non-empty open set includes 0.
*Quite generally, a topological space ''X'' will be T<sub>0</sub> if and only if the [[specialization preorder]] on ''X'' is a [[partial order]]. However, ''X'' will be T<sub>1</sub> if and only if the order is discrete (i.e. agrees with equality). So a space will be T<sub>0</sub> but not T<sub>1</sub> if and only if the specialization preorder on ''X'' is a non-discrete partial order.