On the cardinality of Hausdorff spaces and H-closed spaces

2016/10/28 by Carlson, Nathan, Porter, Jack
#54A25 #54D10 #54D20 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1610.09245

Abstract

We introduce the cardinal invariant aL^′(X) and show that |X|≤ 2^{aL^′(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)≤ aL_c(X). Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound 2^L(X)χ(X) from which it follows that |X|≤ 2^ψ_c(X) if X is H-closed (Dow/Porter [5]). The invariant aL^′(X) is constructed using convergent open ultrafilters and an operator c:\scrP(X)→\scrP(X) with the property clA⊆ c(A)⊆ cl_θ(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 [10] 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|≤ 2^κ, 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 c.

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