A bound for the density of any Hausdorff space

A bound for the density of any Hausdorff space Carlson

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A bound for the density of any Hausdorff space 2024 Topology and its Applications 10.1016/j.topol.2024.108838 carlson2024-a-bound-for-the-density-of-any-hau carlson2024-a-bound-for-the-density-of-any-hau.txt, the project's staged copy 770fefdf9fd0c7c71f91bf6e890dd7f2dc65d791 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. density bounds for Hausdorff 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 The nonquasiregularity degree simultaneously generalizes quasiregularity and the closed pseudocharacter, so a single inequality in nq(X) specialises to both Šapirovskiĭ's regular-space bound and Sun's Hausdorff bound. Quotation withheld because this record is not cleared for redistribution. The new cardinal function nq(X) is not defined for every topological space, so the main bound does not apply universally. Quotation withheld because this record is not cleared for redistribution. A compact T1 space is exhibited for which the invariant fails to exist. Quotation withheld because this record is not cleared for redistribution. The nq(X) factor cannot simply be dropped: examples are given where the plain bound fails. Quotation withheld because this record is not cleared for redistribution. The two classical bounds this paper unifies had complicated proofs that were quite different from one another, which is the gap the paper exists to close. what this limits: the relation between Sapirovskii's density bound and Sun's cardinality bound before this paper: Sun's proof is described as a complicated closing-off argument quite different from the proof of the density bound, so the two results were not visibly connected Quotation withheld because this record is not cleared for redistribution. The class of spaces the theorem covers is named after the invariant being defined on them, so membership is settled by whether the quantity exists rather than by a topological property. what this limits: the scope of the main theorem: it applies to nq-spaces, and a space is an nq-space precisely when nq(X) is defined, which is why a compact T1 example without the invariant falls outside it Quotation withheld because this record is not cleared for redistribution. density density Quotation withheld because this record is not cleared for redistribution. quasiregular quasiregular Quotation withheld because this record is not cleared for redistribution. nonquasiregularity degree nonquasiregularity degree Quotation withheld because this record is not cleared for redistribution. nq-space nq-space Quotation withheld because this record is not cleared for redistribution. Hausdorff space Hausdorff space Quotation withheld because this record is not cleared for redistribution. nonregularity - AS A DEGREE RATHER THAN A DICHOTOMY: instead of a space being regular or not, the paper introduces a cardinal measuring HOW nonquasiregular it is, and the classical bound then holds for every space with that degree as a factor Quotation withheld because this record is not cleared for redistribution. quasiregular - THE HYPOTHESIS WEAKENED ONCE BEFORE IT WAS MEASURED: the classical result was stated for regular spaces, then observed to need only quasiregularity and not even Hausdorff, and only afterwards replaced by a degree Quotation withheld because this record is not cleared for redistribution. the degree - SMALL FOR THE SPACES THAT MATTER: nq(X) is countable when the space is quasiregular and bounded by the closed pseudocharacter when it is Hausdorff, which is what stops the new factor from making the bound vacuous Quotation withheld because this record is not cleared for redistribution. extends → Šapirovskiĭ's density bound d(X) ≤ πχ(X)c(X) for regular spaces Quotation withheld because this record is not cleared for redistribution. extends → Sun's bound πχ(X)c(X)ψc(X) for the cardinality of a Hausdorff space Quotation withheld because this record is not cleared for redistribution. replicates → Šapirovskiĭ's and Sun's theorems (re-derived from one argument) Quotation withheld because this record is not cleared for redistribution. Hausdorff space —is-a→ nq-space Hausdorff space nq-space strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. quasiregular —is-a→ nq-space quasiregular nq-space strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. The density of any space for which the nonquasiregularity degree is defined — in particular every Hausdorff space and every quasiregular space — is at most πχ(X) times c(X) times nq(X). Quotation withheld because this record is not cleared for redistribution. In a Hausdorff space the new cardinal invariant stays small, being bounded above by the closed pseudocharacter. Quotation withheld because this record is not cleared for redistribution. The bound yields a unified proof of Šapirovskiĭ's density bound for regular spaces and Sun's cardinality bound πχ(X)c(X)ψc(X) for Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. It follows that the cardinality of any Hausdorff space is at most πχ(X)c(X)nq(X)wψc(X), which improves Sun's bound because wψc(X) never exceeds ψc(X) in Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. The number of regular open sets of an nq-space is likewise bounded by πχ(X)c(X)nq(X). Quotation withheld because this record is not cleared for redistribution. In a Hausdorff space the nonquasiregularity degree is bounded above by the Lindelöf number. Quotation withheld because this record is not cleared for redistribution. Every Lindelöf Hausdorff space satisfies d(X) ≤ πχ(X)c(X). Quotation withheld because this record is not cleared for redistribution. Two spaces are exhibited, one of them Urysohn, whose density strictly exceeds πχ(X)c(X), showing the extra factor nq(X) cannot be dropped from the bound. Quotation withheld because this record is not cleared for redistribution. The nonquasiregularity degree is not defined for every space: there is a compact T1 space with no nq(X). Quotation withheld because this record is not cleared for redistribution. density density Quotation withheld because this record is not cleared for redistribution. quasiregular space quasiregular space Quotation withheld because this record is not cleared for redistribution. nonquasiregularity degree nq(X) nonquasiregularity degree nq(X) about this quote: THE QUOTE STOPS BEFORE A FORMULA: the set-theoretic condition that follows 'contains a' is garbled in the extraction ('V ∈V V ⊆ nonempty Gκ-set G = U …'), so the quote ends on the word that introduces it. Quotation withheld because this record is not cleared for redistribution. nq-space nq-space Quotation withheld because this record is not cleared for redistribution. weak closed pseudobase weak closed pseudobase 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. regular open set regular open set Quotation withheld because this record is not cleared for redistribution. nq(X) = ℵ0 — For a quasiregular space the nonquasiregularity degree takes its least possible value, the first infinite cardinal — i.e. quasiregular spaces are exactly the 'countably nonquasiregular' ones. Quotation withheld because this record is not cleared for redistribution. χ(X) = 2κ and d(X) > cf (22κ ) > 2κ — In van Douwen's example the character equals 2^κ while the density strictly exceeds 2^κ, so density outruns πχ(X)c(X) and the nq factor is unavoidable. Quotation withheld because this record is not cleared for redistribution.

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