New bounds on the cardinality of Hausdorff spaces and regular spaces

New bounds on the cardinality of Hausdorff spaces and regular spaces Carlson

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New bounds on the cardinality of Hausdorff spaces and regular spaces 2023 Acta Mathematica Hungarica 10.1007/s10474-023-01331-9 carlson2023-new-bounds-on-the-cardinality-of-h carlson2023-new-bounds-on-the-cardinality-of-h.txt, the project's staged copy 65986b6add8534d26a19a95e43c4643bbfc9482a open-access set-theoretic-topology

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. cardinality bounds for Hausdorff and regular spaces Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms bounding |X| by the family of regular open sets and then bounding that family by a closing-off argument on density, so the pseudocharacter can be weakened without loss Quotation withheld because this record is not cleared for redistribution. mechanisms constructing an injection of the space into a family of small subsets indexed by weak closed pseudobases, which transfers existing bounds from ψc(X) to wψc(X) Quotation withheld because this record is not cleared for redistribution. Whether the wψc-based bound holds for compact homogeneous Hausdorff spaces is left unresolved by the paper's own examples. Quotation withheld because this record is not cleared for redistribution. The corresponding bound for general (non-quasiregular) Hausdorff spaces is posed as an open question rather than proved. Quotation withheld because this record is not cleared for redistribution. No separation axiom is assumed anywhere, so every result's hypotheses have to be read locally rather than inherited from the word 'space'. what this limits: any theorem quoted from this paper as being about 'a space': the author makes no global assumption of any separation axiom, so the hypotheses stated in each individual result are the only ones in force Quotation withheld because this record is not cleared for redistribution. The novelty claim for the central new bound is hedged, and the hedge is the author's. what this limits: the priority of the bound on cardinality by density and weak closed pseudocharacter: the author says it appears to be new in the literature rather than asserting it is, and notes a related result was already known for regular spaces Quotation withheld because this record is not cleared for redistribution. Two theorems reach the same kind of conclusion by different sophisticated arguments and the author cannot unify them. what this limits: treating the two density bounds here as one result: the author states the proofs are different sophisticated closing-off arguments and asks, as an open question, whether a common proof exists Quotation withheld because this record is not cleared for redistribution. The substitution the paper's new functions permit is shown not to extend to the compact case. what this limits: generalising the replacement of pseudocharacter by the new weaker functions: compact examples are given showing the substitution fails in the classical bound for the cardinality of a compact Hausdorff space Quotation withheld because this record is not cleared for redistribution. separation axiom separation axiom Quotation withheld because this record is not cleared for redistribution. closing-off argument closing-off argument Quotation withheld because this record is not cleared for redistribution. difficult difficult Quotation withheld because this record is not cleared for redistribution. pseudocharacter pseudocharacter Quotation withheld because this record is not cleared for redistribution. cardinal function cardinal function Quotation withheld because this record is not cleared for redistribution. 'space' - WITH NO SEPARATION AXIOM ATTACHED: in much of the topology literature the bare word silently carries Hausdorff or regular or T1 depending on the author, so this paper states that it assumes none of them globally and every hypothesis is local to its theorem Quotation withheld because this record is not cleared for redistribution. 'difficult' - A CLASSIFICATION OF PROOF TECHNIQUE, ATTRIBUTED: the word is scare-quoted and referred to a particular author and section, and what earns it is that the proofs rely on sophisticated closing-off arguments, not that the statements are hard to understand Quotation withheld because this record is not cleared for redistribution. pseudocharacter and closed pseudocharacter - THE SAME NUMBER ONLY UNDER A HYPOTHESIS: they are ordered for any Hausdorff space and become equal when the space is regular, so whether a bound stated with one is a bound stated with the other depends on a separation axiom Quotation withheld because this record is not cleared for redistribution. extends → the cardinality bounds of Šapirovskiĭ [22] and Sun [23] Quotation withheld because this record is not cleared for redistribution. extends → Gotchev, Tkachenko and Tkachuk's bound πw(X)ot(X)ψc(X) [13] Quotation withheld because this record is not cleared for redistribution. extends → the Bella-Cammaroto bound d(X)t(X)ψc(X) [1] as improved by the author in [6] Quotation withheld because this record is not cleared for redistribution. cardinal function —contrasted-with→ pseudocharacter cardinal function pseudocharacter strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. For a quasiregular Hausdorff space, |X| ≤ πχ(X)^(c(X)wψc(X)). Quotation withheld because this record is not cleared for redistribution. For any Hausdorff space, |X| ≤ πχ(X)^(c(X)dψc(X)wψc(X)). Quotation withheld because this record is not cleared for redistribution. Every Hausdorff space satisfies |X| ≤ 2^(d(X)wψc(X)). Quotation withheld because this record is not cleared for redistribution. The density of any Hausdorff space is at most πχ(X)^(c(X)dψc(X)). Quotation withheld because this record is not cleared for redistribution. The cardinality of a Hausdorff space is at most |RO(X)| raised to wψc(X). Quotation withheld because this record is not cleared for redistribution. Compact examples show that ψ(X) cannot be replaced by dψc(X)wψc(X) in the bound 2^ψ(X) for compact Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. A quasiregular Hausdorff space whose cellularity is at most the continuum and whose π-character is countable has at most 2^c points. Quotation withheld because this record is not cleared for redistribution. For a Hausdorff space, |X| ≤ d(X)^(wt(X)wψc(X)). Quotation withheld because this record is not cleared for redistribution. In a homogeneous Hausdorff space the new dense closed pseudocharacter collapses to the ordinary closed pseudocharacter, so it yields no improvement there. Quotation withheld because this record is not cleared for redistribution. 2^wψc(X) is not a cardinality bound for all σ-compact homogeneous spaces. Quotation withheld because this record is not cleared for redistribution. weak closed pseudocharacter wψc(X) weak closed pseudocharacter wψc(X) Quotation withheld because this record is not cleared for redistribution. dense closed pseudocharacter dψc(X) dense closed pseudocharacter dψc(X) Quotation withheld because this record is not cleared for redistribution. quasiregular space quasiregular space Quotation withheld because this record is not cleared for redistribution. homogeneous space homogeneous space Quotation withheld because this record is not cleared for redistribution. C-saturated subset C-saturated subset Quotation withheld because this record is not cleared for redistribution. weak tightness wt(X) weak tightness wt(X) Quotation withheld because this record is not cleared for redistribution. regular open set regular open set Quotation withheld because this record is not cleared for redistribution.

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