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 T_{3.5} 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
T_{1} spaces at all, but give as an exercise the fact
that points in a T_{2} space are closed.

The terminology T_{0}, T_{1}, T_{2}, etc., follows
the literature, but the student is cautioned
that the literature varies slightly
on its definitions of T_{3} and T_{3.5};
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 T_{0} 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
``R_{0} spaces'' and ``R_{1} 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).