Almost compact subspaces of hyperextensions (Q1333062)

Given a space \(X\) together with an open subbase \(S\) for its topology, the author earlier constructed and studied a superspace \(h(X,S)\) [Period. Math. Hung. 1, 55-80 (1971; Zbl 0219.54021)]. His construction can be considered a generalization of the one used by \textit{J. de Groot} to obtain superextensions [Contrib. Extens. Theory topol. Struct., Proc. Sympos. Berlin 1967, 89-90 (1969; Zbl 0191.212)]. Subspaces of this large superspace (\(X\) typically is not dense in \(h(X,S)\)) were taken up in [the author, Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 15(1972), 111-119 (1973; Zbl 0245.54002)]. The present paper deals again with subspaces of \(h(X,S)\), this time in relation to almost (super- )compactness (\(X\) is almost supercompact relative to \(S\) iff every open cover of \(X\) with members of \(X\) has two members whose union is dense in \(X\); and \(X\) is almost compact if every open cover of \(X\) has a finite subcollection with dense union). In particular, a subspace \(a(X,S)\) is selected which turns out to be ultradense in \(h(X,S)\) and which is important for supercompactness properties of \(X\). Several other subspaces of \(h(X,S)\) are likewise studied. This leads to generalizations of \(H\)- closed extensions as defined by \textit{J. Flachsmeyer} [Math. Z. 94, 349- 381 (1966; Zbl 0147.415)] and generalized by \textit{K. Császár} [Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 19(1976), 49-61 (1977; Zbl 0327.54016)].