Chapter 16: Separation and Regularity Axioms

16.1. Preview. This chapter considers conditions under which points and/or closed sets can be separated by open sets and/or continuous functions. The chart below shows the relations between the main separation conditions. The chart has implications downward; for instance, every T3 space is also a T2 space, and every preregular space is also a symmetric space.

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The two entries in each row of the chart are closely related: a space satisfies the condition in the left column if and only if the space is Kolmogorov and satisfies the condition in the right column in the same row. For instance, a topological space is Hausdorff if and only if it is both Kolmogorov and preregular. We can move from the right column to the left column by taking the Kolmogorov quotient of a space, as in 16.5.

Normality is an interesting condition by itself --- we shall introduce it in 16.26 --- but it fits in the chart only in conjunction with the symmetry condition, since normal by itself does not imply completely regular.

The long list of conditions may be daunting to beginners, or it may seem like hair-splitting to some readers. The student will find it helpful to concentrate on pseudometrizable spaces and completely regular spaces; those will play the greatest role in this book. Most spaces arising in applications in analysis are at least T3.5 spaces, but the abstract theory can be developed more clearly if we classify properties according to the various axioms in the chart. It is possible to decrease the emphasis on some of these properties, but it is not possible to omit them altogether. For instance, some textbooks omit mentioning T1 spaces at all, but give as an exercise the fact that points in a T2 space are closed.

The terminology T0, T1, T2, etc., follows the literature, but the student is cautioned that the literature varies slightly on its definitions of T3 and T3.5; some mathematicians interchange some of the terms in the two columns in our chart. Even mathematicians who agree to our terminology may use it in different ways; for instance, the phrases ``Tychonov space'', ``completely regular T0 space'', and ``completely regular Hausdorff space'' are used interchangeably in the literature; they all describe the same thing.

Most of the separation and regularity axioms are well known, and can be found in many topology books. However, ``symmetric spaces'' and ``preregular spaces'' are not so well known. They are the same (respectively) as the ``R0 spaces'' and ``R1 spaces'' introduced by Davis [1961]. Symmetric spaces were introduced earlier by Shanin [1943]. This book owes a debt to the book of Murdeshwar [1983], which investigates those spaces systematically.

Our choice of emphases is determined by the needs of later chapters. For instance, many topology books concern themselves solely with Hausdorff spaces. This book considers the non-Hausdorff case as well, because one of the best ways to describe a weak topology on a topological vector space (generally Hausdorff, in applications) is as the supremum of a collection of pseudometric topologies (each of which is not Hausdorff).