Wallman compactifications and Wallman realcompactifications

Abstract

In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

Round subsets of Wallman-type compactifications

“…The last part is clear. and Gagrat and Naimpally (1973).) We further define, for each p G w{2£), M" = {Z £ %: p G Z} and 6" = use, available at https://www.cambridge.org/core/terms.…”

“…Now, it is easy to show that if 9 U 9 are two 2£ -filters and 9 X C9, then 9(f)Cfl(f,). Moreover, since 5 = {Z:ZG3f} is a normal base for closed subsets in w(S'), (see , Gagrat and Naimpally (1973) and Njastad (1966)), each closed subset A of <jt (2£) is of the form 6(9) for some 3t-filter. Namely, 9 = {Z G 3f: A C Z} which clearly is a 2C-filter and 0(&) = A.…”

“…Recently, the Wallmantype (real-) compactification <w(3T), (v(2£) respectively) has been studied. (See Alo and Shapiro (1968), Gagrat and Naimpally (1973), Njastad (1966), Steiner and Steiner (1970), Su (1975).) Mandelker (1969) studied the round z -filters and round subsets of jSX.…”

“…A normal base 3E of a space X is a base for closed subsets of X which satisfies the following conditions: (i) 3£ is a ring (i.e., closed under finite unions and intersections), (ii) 2f is [2] Wallman-type 225 disjunctive (i.e., if x is not contained in the closed subset A of X, then there is a Z E 2£ such that xEZCX-A), (iii) 2t is normal (i.e., A, B E 2£ and A n B = 0 , then there exist sets C,DE3? such that A C X -C , BcX-D and C U D = X (See , Gagrat and Naimpally (1973), Njastad (1966), Steiner and Steiner (1970) and Su (1975).) Let 2T be a family of closed subsets of X.…”

“…Let % be a normal base and let <o(2£) be the set of all 2T-ultra-filters. a>{2£) with topology defined as usual (see and Gagrat and Naimpally (1973)) is called a Wallmantype compactification. If in addition 2£ is an intersection ring then v{2£) denotes the subspace of «(3f) which consists of all ST-ultrafilters with the countable intersection property and v{3£) is called a Wallman-type real-compactification.…”