A worked preservation table for subspaces and products of the standard, cofinite, and ordinal topologies

The discrete topology is $T_1$, Hausdorff, regular, normal, completely regular, and completely normal: singleton complements and singleton sets give all required separating opens, and every function to $[0,1]$ is continuous. Its subspaces and products keep the positive properties listed in $T_0$, $T_1$, $T_2$, regularity, $T_3$, complete regularity, and Tychonoffness are hereditary and $T_0$, $T_1$, $T_2$, regularity, $T_3$, complete regularity, and Tychonoffness are productive.

For an infinite set with the cofinite topology of The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, every singleton is closed, so the space is $T_1$; two nonempty open sets intersect, so it is not Hausdorff. This displays why the positive table does not infer a higher axiom from $T_1$ alone. An ordinal with the order topology of The order topology on an ordinal, with the half-open intervals $(\alpha, \beta]$ and the initial segments $[0, \beta]$ as a basis supplies the contrasting clopen order intervals used by the deleted-plank construction.