Separation Axiom - Relationships Between The Axioms

Relationships Between The Axioms

The T₀ axiom is special in that it can be not only added to a property (so that completely regular plus T₀ is Tychonoff) but also 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 below:

In this table, you go from the right side to the left side by adding the requirement of T₀, and you go from the left side to the right side by removing that requirement, using the Kolmogorov quotient operation. (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 following diagram:

In this diagram, the non-T₀ version of a condition is on the left side of the slash, and the T₀ version is on the right side. Letters are used for abbreviation as follows: "P" = "perfectly", "C" = "completely", "N" = "normal", and "R" (without a subscript) = "regular". A bullet indicates that there is no special name for a space at that spot. The dash at the bottom indicates no condition.

You can combine two properties using this diagram by following the diagram upwards until both branches meet. For example, if a space is both completely normal ("CN") and completely Hausdorff ("CT₂"), then following both branches up, you find the spot "•/T₅". Since completely Hausdorff spaces are T₀ (even though completely normal spaces may not be), you take the T₀ side of the slash, so a completely normal completely Hausdorff space is the same as a T₅ space (less ambiguously known as a completely normal Hausdorff space, as you can see in the table above).

As you can see from the diagram, normal and R₀ together imply a host of other properties, since combining the two properties leads you to follow a path through the many nodes on the rightside branch. Since regularity is the most well known of these, spaces that are both normal and R₀ are typically called "normal regular spaces". In a somewhat similar fashion, spaces that are both normal and T₁ are often called "normal Hausdorff spaces" by people that wish to avoid the ambiguous "T" notation. These conventions can be generalised to other regular spaces and Hausdorff spaces.