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# History of the separation axioms

In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept.

 Contents

## 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, T (or Tπ), and T5.

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 was not necessarily true! It depended on precisely how the definitions were phrased. For example, a regular space (called T3) does not have to be a Hausdorff space (called T2), at least not if you use the simplest definition of regular space.

## Different definitions

Every author agreed on T0, T1, and T2. But for the other axioms, different authors could use radically different definitions, depending on what they were working upon. The reason why these differences could develop is that, if you assume that a topological space satisfies the T1 axiom, then the various definitions were (almost always) equivalent. Thus if you were going to make that assumption, then you would want to use the simplest definition. But if you 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 relation between Ti and Tj, 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 Ti. 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 Kelley in 1955, usually did not assume T1, so they studied the separation axioms in the greatest generality from the beginning. Thus they used the more complicated definitions for Ti, so that they would always have a nice property relating Ti to Tj. 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.

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.

## Regularity axioms

Kolmogorov quotients, preregular spaces, and the Ri sequence

## Completely Hausdorff spaces

T, the meaning of "completely", and Willard's switch

03-10-2013 05:06:04