Separation axiom

The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties.

The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points.

Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways).

Points x and y will be considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by a function, if and only if their singleton sets {x} and {y} are separated according to the corresponding criterion.

The T0 axiom is special in that it can not only be added to a property (so that completely regular plus T0 is Tychonoff) but also be subtracted from a property (so that Hausdorff minus T0 is R1), in a fairly precise sense; see Kolmogorov quotient for more information.

When applied to the separation axioms, this leads to the relationships in the table to the left below.

(The names in parentheses given on the left side of this table are generally ambiguous or at least less well known; but they are used in the diagram below.)

Other than the inclusion or exclusion of T0, the relationships between the separation axioms are indicated in the diagram to the right.

Letters are used for abbreviation as follows: "P" = "perfectly", "C" = "completely", "N" = "normal", and "R" (without a subscript) = "regular".

[NB: This diagram does not reflect that perfectly normal spaces are always regular; the editors are working on this now.]

Illustrations of the properties of Hausdorffness, regularity and normality
An illustration of some of the separation axioms. Grey amorphous broken-outline regions indicate open neighbourhoods of disjoint closed sets or points: red solid-outline circles denote closed sets while black dots represent points.
Hasse diagram of the separation axioms.
Hasse diagram of the separation axioms.