A class of separation axioms
S(α), one for each ordinal α > 0, is introduced. Axiom S(1) is the Hausdorff
property, S(2) is Urysohn and regular implies S(w0), where w0 is the first infinite
ordinal. Minimal S(α) and S(α)-closed spaces are characterized, and many of the
known results for minimal S(i) and S(i)-closed are extended, i = 1,2. For a limit
ordinal α > 0, minimal ∕S(α) spaces are shown to be regular. A new approach to the
study of minimal regular spaces is provided by showing that the properties of
minimal regular and minimal S(w0) are equivalent even though the concepts of
regularity and S(w0) are not equivalent.
A new subclass of regular spaces-called OCE-regular spaces-is introduced and
used to develop an extension theory for regular spaces, which subsumes the
regular-closed extension theory developed by Harris. It is proven that every regular
space can be densely embedded in an OCE-regular space and that the set of
OCE-regular extensions of a regular space is in a one-to-one correspondence
with a set of generalized Smirnov proximities compatible with the regular
space.
