Every simply ordered set is one of these spaces in the ordered topology.
The Smirnov metrization theorem asserts that a locally metrizable space must be a paracompact one of these in order to be metrizable. The real numbers in a topology with basis open intervals and open intervals minus the set of inverse positive integers is not regular, but it is this. Every simply ordered set is one of these spaces in the ordered topology. A space X is this if and only if the diagonal is closed in the product space X times X. In a space of this type, a sequence of points converges to at most one point. This class of topological space has the property that any two points have disjoint neighborhoods and thus possess finite subsets that are closed by the T2 separation axiom. For 10 points, what is this type of topological space, named for a mathematician also known for a concept of dimension often used for fractals?
The Smirnov metrization theorem asserts that a locally metrizable space must be a paracompact one of these in order to be metrizable.
Hausdorff space [or T2 space before mentioned]