Arhangel'skiĭ's solution to Alexandroff's problem: A survey

Abstract

In 1969, Arhangel'skiȋ proved that |X| 2 χ(X)L(X) for every Hausdorff space X. This beautiful inequality solved a nearly fifty-year old question raised by Alexandroff and Urysohn. In this paper we survey a wide range of generalizations and variations of Arhangel'skiȋ's inequality. We also discuss open problems and an important legacy of the theorem: the emergence of the closure method as a fundamental unifying device in cardinal functions.

On the cardinality of Hausdorff spaces and H-closed spaces

“…The Dow-Porter result given in Corollary 4.6 now follows, using a proof similar to the proof of that corollary. We see then two very different proofs of this result, one using open ultrafilters (which generalizes to a result for all Hausdorff spaces, Theorem 4.4) and the other using κ-nets [10] which can be reframed in terms of κ-filters as in Theorem 3.11. We note that in [15], Porter used a different type of open ultrafilter approach.…”

“…The filter characterization of H-closed spaces used in Theorem 2.21 using c-adherence of a filter in conjunction with a variation of a method used by Hodel [10] for nets provides a direct path for proving that the cardinality of an H-closed space X is bounded by 2 ψc(X) . Suppose now that X is H-closed and let V be an open cover of X.…”

“…The invariant aL ′ (X) is constructed using convergent open ultrafilters and an operator c : P(X) → P(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.…”

“…In §3 we use a filter characterization of H-closed spaces given in Theorem 2.21 and the c-adherence of a filter to give another proof that the cardinality of an H-closed space X is bounded by 2 ψc(X) . This particular method can be seen as a variation of the method used by Hodel [10] for nets. We present two examples at the end of §3.…”

“…For all undefined notions see Engelking [6], Juhász [12], or Porter-Woods [17]. Hodel's survey paper [10] also contains thorough discussion of many cardinal invariants and cardinality bounds related to those discussed in this study.…”