phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | density bounds for Hausdorff spaces— “The density of a topological space X, denoted by d(X), is the least cardinality of a dense subset of X.” | |
| claim-type | existence— “We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree nq(X), which is countable when X is quasiregular, and showing d(X) ≤ πχ(X)c(X)nq(X) for any Hausdorff space X.” | |
| method — evidence | — “We prove now the central result in this paper.” | |
| mechanism | 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.— “This shows that the nonquasiregular degree generalizes both quasiregularity in spaces for which nq(X) is defined and the closed pseudocharacter in Hausdorff spaces.” | mechanisms[0] |
| extends | Šapirovskiĭ's density bound d(X) ≤ πχ(X)c(X) for regular spaces— “We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree nq(X), which is countable when X is quasiregular, and showing d(X) ≤ πχ(X)c(X)nq(X) for any Hausdorff space X.” | claim_relations[0] |
| extends | Sun's bound πχ(X)c(X)ψc(X) for the cardinality of a Hausdorff space— “This improves Sun’s theorem and shows that the degree to which a space is nonquasiregular plays an important role in the cardinality of a Hausdorff space.” | claim_relations[1] |
| replicates | Šapirovskiĭ's and Sun's theorems (re-derived from one argument)— “This results in a unified proof of ˇSapirovski˘ı’s and Sun’s theorems, gives an improved bound for the cardinality of a Hausdorff space, and answers Question 3.3 in [2].” | claim_relations[2] |
| sense | 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— “we introduce the nonquasiregular degree nq(X) (Definition 2.2) which gives a “measure” of how nonquasiregular a space is.” | senses[0] |
| sense | 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 | senses[1] |
| sense | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[2] |
| edge | Hausdorff space —is-a→ nq-space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[0] |
| edge | quasiregular —is-a→ nq-space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| for which use | limitation — 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| for which use | limitation — 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
| replication | untested | |
| limitation | The new cardinal function nq(X) is not defined for every topological space, so the main bound does not apply universally. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | A compact T1 space is exhibited for which the invariant fails to exist. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | The nq(X) factor cannot simply be dropped: examples are given where the plain bound fails. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| limitation | 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. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| limitation | 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. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | In a Hausdorff space the new cardinal invariant stays small, being bounded above by the closed pseudocharacter. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| finding | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[2] |
| finding | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[3] |
| finding | The number of regular open sets of an nq-space is likewise bounded by πχ(X)c(X)nq(X). Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[4] |
| finding | In a Hausdorff space the nonquasiregularity degree is bounded above by the Lindelöf number. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[5] |
| finding | Every Lindelöf Hausdorff space satisfies d(X) ≤ πχ(X)c(X). Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[6] |
| finding | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[7] |
| finding | The nonquasiregularity degree is not defined for every space: there is a compact T1 space with no nq(X). Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[8] |
| quantity | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[0] |
| quantity | 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 — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.quantities[1] |
| definition | density Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| definition | quasiregular space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| definition | nonquasiregularity degree nq(X) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[2] |
| evidence status note | definition — 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. | content_tags.definitions[2] |
| definition | nq-space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[3] |
| definition | weak closed pseudobase Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[4] |
| definition | weak closed pseudocharacter wψc(X) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[5] |
| definition | dense closed pseudocharacter dψc(X) Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[6] |
| definition | regular open set Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[7] |
| concept | density Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[0] |
| concept | quasiregular Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[1] |
| concept | nonquasiregularity degree Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[2] |
| concept | nq-space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[3] |
| concept | Hausdorff space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[4] |
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