standOff, and the article is at its publisher via the link below. When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.| relation to other work | claim_relations | how this paper stands to another: agrees, contradicts, extends, qualifies |
| claim type | claim_type | the shape of the claim: existence, mechanism, method, critique, prediction, null |
| concept edge | concept_edges | an asserted relation between two concepts, with the sentence asserting it |
| concept | concepts | an idea the paper works with, quoted where the paper uses it |
| content tag | content_tags | a finding, definition, quantity, statistic or variable named in the paper |
| limitation | limitations | what the paper says its evidence cannot support |
| mechanism | mechanisms | how the reported effect is said to come about |
| method | method | how the paper produced its evidence |
| phenomenon | phenomenon | the effect or regularity under study |
| sense | senses | a distinct use of a term, quoted where the paper fixes it |
Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 2 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.
This record holds no controls. A layer is empty when the tagger recorded nothing in it — which is not the same as the paper having nothing to record.
| facet | tag | field | conf. | whose words |
|---|---|---|---|---|
| claim_type | existence Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | claim_type | the paper's words | |
| phenomenon | density bounds for Hausdorff spaces Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | phenomenon | the paper's words | |
| method | mathematical-proof Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mathematical-proof | method | the paper's words | |
| mechanisms | mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | mechanisms[0] | the tagger's words | |
| limitations | The new cardinal function nq(X) is not defined for every topological space, so the main bound does not apply universally. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | limitations[0] | the tagger's words | |
| limitations | A compact T1 space is exhibited for which the invariant fails to exist. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | limitations[1] | the tagger's words | |
| limitations | The nq(X) factor cannot simply be dropped: examples are given where the plain bound fails. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | limitations[2] | the tagger's words | |
| limitations | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[3] | 1.0 | the tagger's words |
| limitations | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[4] | 1.0 | the tagger's words |
| concepts | density Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: density | concepts[0] | the paper's words | |
| concepts | quasiregular Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: quasiregular | concepts[1] | the paper's words | |
| concepts | nonquasiregularity degree Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: nonquasiregularity degree | concepts[2] | the paper's words | |
| concepts | nq-space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: nq-space | concepts[3] | the paper's words | |
| concepts | Hausdorff space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Hausdorff space | concepts[4] | the paper's words | |
| senses | 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 — this record is not cleared for redistribution. The sentence is in the paper at the link above. | senses[0] | 1.0 | the tagger's words |
| senses | 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 — this record is not cleared for redistribution. The sentence is in the paper at the link above. | senses[1] | 1.0 | the tagger's words |
| senses | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | senses[2] | 1.0 | the tagger's words |
| claim_relations | extends → Šapirovskiĭ's density bound d(X) ≤ πχ(X)c(X) for regular spaces Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | claim_relations[0] | the tagger's words | |
| claim_relations | extends → Sun's bound πχ(X)c(X)ψc(X) for the cardinality of a Hausdorff space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | claim_relations[1] | the tagger's words | |
| claim_relations | replicates → Šapirovskiĭ's and Sun's theorems (re-derived from one argument) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | claim_relations[2] | the tagger's words | |
| concept_edges | Hausdorff space —is-a→ nq-space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: Hausdorff space · nq-space | concept_edges[0] | the tagger's words | |
| concept_edges | quasiregular —is-a→ nq-space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: quasiregular · nq-space | concept_edges[1] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[0] | the tagger's words | |
| content_tags | In a Hausdorff space the new cardinal invariant stays small, being bounded above by the closed pseudocharacter. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[1] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[2] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[3] | the tagger's words | |
| content_tags | The number of regular open sets of an nq-space is likewise bounded by πχ(X)c(X)nq(X). Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[4] | the tagger's words | |
| content_tags | In a Hausdorff space the nonquasiregularity degree is bounded above by the Lindelöf number. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[5] | the tagger's words | |
| content_tags | Every Lindelöf Hausdorff space satisfies d(X) ≤ πχ(X)c(X). Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[6] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[7] | the tagger's words | |
| content_tags | The nonquasiregularity degree is not defined for every space: there is a compact T1 space with no nq(X). Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[8] | the tagger's words | |
| content_tags | density Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: density | content_tags.definitions[0] | the tagger's words | |
| content_tags | quasiregular space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: quasiregular space | content_tags.definitions[1] | the tagger's words | |
| content_tags | nonquasiregularity degree nq(X) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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. also in: nonquasiregularity degree nq(X) | content_tags.definitions[2] | the tagger's words | |
| content_tags | nq-space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: nq-space | content_tags.definitions[3] | the tagger's words | |
| content_tags | weak closed pseudobase Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: weak closed pseudobase | content_tags.definitions[4] | the tagger's words | |
| content_tags | weak closed pseudocharacter wψc(X) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: weak closed pseudocharacter wψc(X) | content_tags.definitions[5] | the tagger's words | |
| content_tags | dense closed pseudocharacter dψc(X) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: dense closed pseudocharacter dψc(X) | content_tags.definitions[6] | the tagger's words | |
| content_tags | regular open set Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: regular open set | content_tags.definitions[7] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.quantities[0] | the tagger's words | |
| content_tags | 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 record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.quantities[1] | the tagger's words |
Tags that could not be placed in the text:
What was deliberately not tagged, and why, is recorded per paper by the tagger. Where a paper has no such record, this page says so rather than implying that nothing was declined.