Definition:Tychonoff Separation Axioms

Definition
The Separation Axioms (sometimes known as the Tychonoff Separation Axioms) are a classification system for topological spaces.

They are not axiomatic as such, but conditions that may or may not apply to general or specific topological spaces.

In general, each condition is stronger than the previous one, with subtleties.

For all of these definitions, $T = \left({X, \vartheta}\right)$ be a topological space with topology $\vartheta$.

Linguistic Note
The letter $T$ comes from the German Trennungsaxiom, which means separation axiom.

Naming Conventions
There are different ways of naming the separation axioms. The technique for this site is to follow the convention used in the Separation axiom page at Wikipedia. The various naming schemes are inconsistent with each other and confusing, and no completely satisfactory convention has been defined. It is suggested that the system used here is more modern than others, but there is little evidence one way or another.

An attempt has been made on the appropriate pages to mention the alternative names of these spaces, but this is inconsistent and possibly inaccurate. The important things to note are the conditions themselves and the relations between them. This is a new area of mathematics in which research is ongoing, and the whole area of ground may shift again completely in the near future.