Vol. 45, No. 1, 1973

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S(α) spaces and regular Hausdorff extensions

Jack Ray Porter and Charles I. Votaw

Vol. 45 (1973), No. 1, 327–345

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.

Mathematical Subject Classification 2000
Primary: 54D15
Received: 29 October 1971
Revised: 7 April 1972
Published: 1 March 1973
Jack Ray Porter
Charles I. Votaw