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Concept
History of the separation axioms
Summary
The history of the separation axioms in general topology has been convoluted, with many meanings competing for the same terms and many terms competing for the same concept.
Before the current general definition of topological space, there were many definitions offered, some of which assumed (what we now think of as) some separation axioms. For example, the definition given by Felix Hausdorff in 1914 is equivalent to the modern definition plus the Hausdorff separation axiom.
The separation axioms, as a group, became important in the study of metrisability: the question of which topological spaces can be given the structure of a metric space. Metric spaces satisfy all of the separation axioms; but in fact, studying spaces that satisfy only some axioms helps build up to the notion of full metrisability.
The separation axioms that were first studied together in this way were the axioms for accessible spaces, Hausdorff spaces, regular spaces, and normal spaces. Topologists assigned these classes of spaces the names T1, T2, T3, and T4. Later this system of numbering was extended to include T0, T21⁄2, T31⁄2 (or Tπ), T5, and T6.
But this sequence had its problems. The idea was supposed to be that every Ti space is a special kind of Tj space if i > j. But this is not necessarily true, as definitions vary. For example, a regular space (called T3) does not have to be a Hausdorff space (called T2), at least not according to the simplest definition of regular spaces.
Every author agreed on T0, T1, and T2. For the other axioms, however, different authors could use significantly different definitions, depending on what they were working on. These differences could develop because, if one assumes that a topological space satisfies the T1 axiom, then the various definitions are (in most cases) equivalent. Thus, if one is going to make that assumption, then one would want to use the simplest definition. But if one did not make that assumption, then the simplest definition might not be the right one for the most useful concept; in any case, it would destroy the (transitive) entailment of Ti by Tj, allowing (for example) non-Hausdorff regular spaces.
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