On the cardinality of Hausdorff spaces and H-closed spaces

We introduce the cardinal invariant aL′(X) and show that |X|≤2aL′(X)χ(X) for any Hausdorff space X (a corollary of Theorem 4.4). This invariant has the properties a) aL′(X)=ℵ0 if X is H-closed, and b) aL(X)≤aL′(X)≤aLc(X). Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound 2L(X)χ(X) from which it follows that |X|≤2ψc(X) if X is H-closed (Dow/Porter [5]). The invariant aL′(X) is constructed using convergent open ultrafilters and an operator cˆ:P(X)→P(X) with the property clA⊆cˆ(A)⊆clθ(A) for all A⊆X. As a comparison with this open ultrafilter approach, in §3 we additionally give a κ-filter variation of Hodel's proof [8] of the Dow-Porter result. Finally, for an infinite cardinal κ, in §5 we introduce κwH-closed spaces, κH′-closed spaces, and κH″-closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if κ is an infinite cardinal and X a κwH-closed space with a dense set of isolated points such that χ(X)≤κ, then |X|≤2κ, and b) if X is κH′-closed or κH″-closed then aL′(X)≤κ. This latter result relates these notions to the invariant aL′(X) and the operator cˆ.