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 WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below.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 An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason.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 1 relation 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 | cardinality bounds for Hausdorff and regular 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 | |
| mechanisms | mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | mechanisms[1] | the tagger's words | |
| limitations | Whether the wψc-based bound holds for compact homogeneous Hausdorff spaces is left unresolved by the paper's own examples. 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 | The corresponding bound for general (non-quasiregular) Hausdorff spaces is posed as an open question rather than proved. 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 | No separation axiom is assumed anywhere, so every result's hypotheses have to be read locally rather than inherited from the word 'space'. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[2] | 1.0 | the tagger's words |
| limitations | The novelty claim for the central new bound is hedged, and the hedge is the author's. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[3] | 1.0 | the tagger's words |
| limitations | Two theorems reach the same kind of conclusion by different sophisticated arguments and the author cannot unify them. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[4] | 1.0 | the tagger's words |
| limitations | The substitution the paper's new functions permit is shown not to extend to the compact case. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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 | limitations[5] | 1.0 | the tagger's words |
| concepts | separation axiom Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: separation axiom | concepts[0] | the paper's words | |
| concepts | closing-off argument Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: closing-off argument | concepts[1] | the paper's words | |
| concepts | difficult Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: difficult | concepts[2] | the paper's words | |
| concepts | pseudocharacter Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: pseudocharacter | concepts[3] | the paper's words | |
| concepts | cardinal function Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: cardinal function | concepts[4] | the paper's words | |
| senses | '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 — 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 | '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 — 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 | 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 — 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 → the cardinality bounds of Šapirovskiĭ [22] and Sun [23] 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 → Gotchev, Tkachenko and Tkachuk's bound πw(X)ot(X)ψc(X) [13] 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 | extends → the Bella-Cammaroto bound d(X)t(X)ψc(X) [1] as improved by the author in [6] 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 | cardinal function —contrasted-with→ pseudocharacter 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: cardinal function · pseudocharacter | concept_edges[1] | the tagger's words | |
| content_tags | For a quasiregular Hausdorff space, |X| ≤ πχ(X)^(c(X)wψc(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 | For any Hausdorff space, |X| ≤ πχ(X)^(c(X)dψc(X)wψc(X)). 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 | Every Hausdorff space satisfies |X| ≤ 2^(d(X)wψc(X)). 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 | The density of any Hausdorff space is at most πχ(X)^(c(X)dψc(X)). 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 cardinality of a Hausdorff space is at most |RO(X)| raised to wψc(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 | 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 — 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 | 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 — 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 | For a Hausdorff space, |X| ≤ d(X)^(wt(X)wψc(X)). 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 | In a homogeneous Hausdorff space the new dense closed pseudocharacter collapses to the ordinary closed pseudocharacter, so it yields no improvement there. 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 | 2^wψc(X) is not a cardinality bound for all σ-compact homogeneous spaces. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. | content_tags.findings[9] | 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[0] | 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[1] | 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[2] | the tagger's words | |
| content_tags | homogeneous space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: homogeneous space | content_tags.definitions[3] | the tagger's words | |
| content_tags | C-saturated subset Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: C-saturated subset | content_tags.definitions[4] | the tagger's words | |
| content_tags | weak tightness wt(X) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: weak tightness wt(X) | content_tags.definitions[5] | 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[6] | 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.