History of the separation axioms

Separation axioms
in topological spaces
Separation axioms; in topological spaces
Kolmogorov classification
T0	(Kolmogorov)
T1	(Fréchet)
T2	(Hausdorff)
T2½	(Urysohn)
completely T2	(completely Hausdorff)
T3	(regular Hausdorff)
T3½	(Tychonoff)
T4	(normal Hausdorff)
T5	(completely normal; Hausdorff)
T6	(perfectly normal; Hausdorff)
	History;

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.

Origins

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, T_2½, T_3½ (or T_π), T₅, and T₆.

But this sequence had its problems. The idea was supposed to be that every T_i space is a special kind of T_j 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.

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 was used as late as 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. But usage is still not consistent.

Completely Hausdorff, Urysohn, and T_21⁄2 spaces

Steen and Seebach define a Urysohn space as "a space with a Urysohn function for any two points". Willard calls this a completely Hausdorff space. Steen & Seebach define a completely Hausdorff space or T_21⁄2 space as a space in which every two points are separated by closed neighborhoods, which Willard calls a Urysohn space or T_21⁄2 space.