History of The Separation Axioms - Different Definitions

Different Definitions

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 T_i by T_j, allowing (for example) non-Hausdorff regular spaces.

Topologists working on the metrisation problem generally did assume T₁; after all, all metric spaces are T₁. Thus, they used the simplest definitions for the T_i. Then, for those occasions when they did not assume T₁, they used words ("regular" and "normal") for the more complicated definitions, in order to contrast them with the simpler ones. This approach reached its zenith in 1970 with the publication of Counterexamples in Topology by Lynn A. Steen and J. Arthur Seebach, Jr.

In contrast, general topologists, led by John L. Kelley in 1955, usually did not assume T₁, so they studied the separation axioms in the greatest generality from the beginning. They used the more complicated definitions for T_i, so that they would always have a nice property relating T_i to T_j. Then, for the simpler definitions, they used words (again, "regular" and "normal"). Both conventions could be said to follow the "original" meanings; the different meanings are the same for T₁ spaces, which was the original context. But the result was that different authors used the various terms in precisely opposite ways. Adding to the confusion, some literature will observe a nice distinction between an axiom and the space that satisfies the axiom, so that a T₃ space might need to satisfy the axioms T₃ and T₀ (e.g., in the Encyclopedic Dictionary of Mathematics, 2nd ed.).

Since 1970, the general topologists' terms have been growing in popularity, including in other branches of mathematics, such as analysis. (Thus we use their terms in Wikipedia.) But usage is still not consistent.