S(α) spaces and regular Hausdorff extensions

Abstract

by showing that the properties of minimal regular and minimal S(WQ) are equivalent even though the concepts of regularity and S(WQ) are not equivalent. A new subclass of regular spaces-called OCl£-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 OCϋv-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. l Introduction* It is well-known that the properties of regularity (includes TO and complete Hausdorff (every pair of distinct points can be separated by a real-valued continuous function) are independent of each other and yet, are implied by completely regularity (includes TΊ) and imply Urysohn (every pair of distinct points can be separated by closed neighborhoods). In §2 a method of distinguishin g between regular and completely Hausdorff is developed by defining a class of separation axioms S(a), one for each ordinal a > 0. This class can be thought of as a measuring rod since it is linearly ordered in the sense that an S(β) space is S(a) if β ^ a. In particular, if wQ denotes the first infinite ordinal and wt the first uncountable ordinal, we prove that a regular space is S(w0) but not necessarily S(w0 + 1) whereas a completely Hausdorff space is S(a) for any ordinal a 0, a minimal S(ά) space is regular, and a space is minimal regular if and only if it is minimal S(w0) even though the concepts of regularity and S(w0) are not equivalent. A space is shown to be regular-closed if and only if the space is regular