phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | cardinality bounds for Hausdorff and regular spaces— “Many bounds on the cardinality of a topological space X involve the pseudocharacter cardinal function, ψ(X), or its variant, the closed pseudocharacter ψc(X).” | |
| claim-type | existence— “Using weaker versions of the cardinal function ψc(X), we derive a series of new bounds for the cardinality of Hausdorff spaces and regular spaces that do not involve ψc(X) nor its variants at all.” | |
| method — evidence | — “As their proofs rely on sophisticated closing-off arguments, these results are deemed “difficult” in the sense of Hodel [15], section 4.” | |
| mechanism | 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— “This is achieved by establishing that |X| ≤ |RO(X)|wψc(X), where RO(X) is the collection of regular open subsets of X, and, using a modified version of Sun’s closing-off argument, that d(X) ≤ πχ(X)c(X)dψc(X) for any Hausdorff space X.” | mechanisms[0] |
| mechanism | 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)— “Using one-to-one map arguments, we can replace ψc(X) with wψc(X) in other cardinality bounds.” | mechanisms[1] |
| extends | the cardinality bounds of Šapirovskiĭ [22] and Sun [23]— “We improve the results of ˇSapirovski˘ı and Sun by showing |X| ≤ πχ(X)c(X)wψc(X) if X is regular, and |X| ≤ πχ(X)c(X)dψc(X)wψc(X) if X is Hausdorff.” | claim_relations[0] |
| extends | Gotchev, Tkachenko and Tkachuk's bound πw(X)ot(X)ψc(X) [13]— “We show in the next result that ψc(X) can be replaced with wψc(X).” | claim_relations[1] |
| extends | the Bella-Cammaroto bound d(X)t(X)ψc(X) [1] as improved by the author in [6]— “We show in Theorem 4.17 that in fact ψc(X) can further be replaced with wψc(X).” | claim_relations[2] |
| sense | '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— “We make no global assumptions of any separation axiom on a topological space in this paper” | senses[0] |
| sense | '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— “As their proofs rely on sophisticated closing-off arguments, these results are deemed “difficult” in the sense of Hodel” | senses[1] |
| sense | 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— “It is clear that ψ(X) ≤ ψc(X) ≤ χ(X) for any Hausdorff space X. Furthermore, if X is regular, it can be seen that ψ(X) = ψc(X)” | senses[2] |
| edge | cardinal function —contrasted-with→ pseudocharacter— “In this study we introduce two cardinal functions for a Hausdorff space X that are weaker than the closed pseudocharacter” | concept_edges[1] |
| for which use | limitation — 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— “We make no global assumptions of any separation axiom on a topological space in this paper” | limitations[2] |
| for which use | limitation — 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— “The latter result appears to be new in the literature even when wψc(X) is replaced with ψc(X), although it was known that |X| ≤ 2d(X)ψ(X) if X is regular” | limitations[3] |
| for which use | limitation — 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— “The proofs are sophisticated closing-off arguments, but different. We ask the following: Question 3.3. Is there a common proof of Theorems 3.1 and 3.2” | limitations[4] |
| for which use | limitation — 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— “Compact examples show that ψ(X) cannot be replaced with dψc(X)wψc(X) in the bound 2ψ(X) for the cardinality of a compact Hausdorff space X” | limitations[5] |
| replication | untested | |
| limitation | Whether the wψc-based bound holds for compact homogeneous Hausdorff spaces is left unresolved by the paper's own examples.— “However, even given Examples 3.11 and 5.1 it is still unclear whether 2wψc(X) is a bound for the cardinality of all compact, homogeneous Hausdorff spaces.” | limitations[0] |
| limitation | The corresponding bound for general (non-quasiregular) Hausdorff spaces is posed as an open question rather than proved.— “Question 4.8. If X is Hausdorff, is |X| ≤ 2c(X)πχ(X)ot(X)?” | limitations[1] |
| limitation | No separation axiom is assumed anywhere, so every result's hypotheses have to be read locally rather than inherited from the word 'space'.— “We make no global assumptions of any separation axiom on a topological space in this paper” | limitations[2] |
| limitation | The novelty claim for the central new bound is hedged, and the hedge is the author's.— “The latter result appears to be new in the literature even when wψc(X) is replaced with ψc(X), although it was known that |X| ≤ 2d(X)ψ(X) if X is regular” | limitations[3] |
| limitation | Two theorems reach the same kind of conclusion by different sophisticated arguments and the author cannot unify them.— “The proofs are sophisticated closing-off arguments, but different. We ask the following: Question 3.3. Is there a common proof of Theorems 3.1 and 3.2” | limitations[4] |
| limitation | The substitution the paper's new functions permit is shown not to extend to the compact case.— “Compact examples show that ψ(X) cannot be replaced with dψc(X)wψc(X) in the bound 2ψ(X) for the cardinality of a compact Hausdorff space X” | limitations[5] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | For a quasiregular Hausdorff space, |X| ≤ πχ(X)^(c(X)wψc(X)).— “Theorem 3.9. If X is quasiregular and Hausdorff then |X| ≤ πχ(X)c(X)wψc(X).” | content_tags.findings[0] |
| finding | For any Hausdorff space, |X| ≤ πχ(X)^(c(X)dψc(X)wψc(X)).— “Theorem 3.10. If X is Hausdorff then |X| ≤ πχ(X)c(X)dψc(X)wψc(X).” | content_tags.findings[1] |
| finding | Every Hausdorff space satisfies |X| ≤ 2^(d(X)wψc(X)).— “Theorem 4.9. If X is Hausdorff then |X| ≤ 2d(X)wψc(X).” | content_tags.findings[2] |
| finding | The density of any Hausdorff space is at most πχ(X)^(c(X)dψc(X)).— “Theorem 3.2. If X is Hausdorff then d(X) ≤ πχ(X)c(X)dψc(X).” | content_tags.findings[3] |
| finding | The cardinality of a Hausdorff space is at most |RO(X)| raised to wψc(X).— “Proposition 3.8. If X is Hausdorff, then |X| ≤ |RO(X)|wψc(X).” | content_tags.findings[4] |
| finding | Compact examples show that ψ(X) cannot be replaced by dψc(X)wψc(X) in the bound 2^ψ(X) for compact Hausdorff spaces.— “Compact examples show that ψ(X) cannot be replaced with dψc(X)wψc(X) in the bound 2ψ(X) for the cardinality of a compact Hausdorff space X.” | content_tags.findings[5] |
| finding | A quasiregular Hausdorff space whose cellularity is at most the continuum and whose π-character is countable has at most 2^c points.— “It follows from Corollary 4.6(3) that a quasiregular Hausdorff space with cellularity at most c and countable π-character has cardinality at most 2c.” | content_tags.findings[6] |
| finding | For a Hausdorff space, |X| ≤ d(X)^(wt(X)wψc(X)).— “Theorem 4.17. If X is Hausdorff then |X| ≤ d(X)wt(X)wψc(X).” | content_tags.findings[7] |
| finding | In a homogeneous Hausdorff space the new dense closed pseudocharacter collapses to the ordinary closed pseudocharacter, so it yields no improvement there.— “Thus if X is homogeneous and Hausdorff we have dψc(X) = ψc(X).” | content_tags.findings[8] |
| finding | 2^wψc(X) is not a cardinality bound for all σ-compact homogeneous spaces.— “Therefore, 2wψc(X) is not a bound for the cardinality of all σ- compact homogeneous spaces.” | content_tags.findings[9] |
| definition | weak closed pseudocharacter wψc(X)— “We define wψc(x, X) to be the least infinite cardinal κ such that x has a weak closed pseudobase of cardinality κ. The weak closed pseudocharacter wψc(X) is defined as wψc(X) = sup{wψc(x, X) : x ∈ X}.” | content_tags.definitions[0] |
| definition | dense closed pseudocharacter dψc(X) | content_tags.definitions[1] |
| definition | quasiregular space | content_tags.definitions[2] |
| definition | homogeneous space | content_tags.definitions[3] |
| definition | C-saturated subset | content_tags.definitions[4] |
| definition | weak tightness wt(X) | content_tags.definitions[5] |
| 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[6] |
| concept | separation axiom 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 | closing-off argument 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 | difficult 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 | 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. | concepts[3] |
| concept | cardinal function 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] |
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.
Tag layer © the Meta-Tagging Project, CC BY 4.0. The paper text is © its authors/publisher
and is linked, not reproduced. Excerpts shown are short quotations for identification and verification.
· back to index
· concepts · authors
· עברית
© 2026 Shir Sivroni · ORCID 0009-0000-2597-2824