Algebraic structures within subsets of Hamel and
      Sierpiński-Zygmund functions

Płotka, Krzysztof

doi:10.36045/bbms/1442364591

Cited by 4 publications

(4 citation statements)

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“…In particular, the problem of the value L for ES \ SES, and related classes, have, lately, attracted the attention of several authors. (See, e.g., [9,25,32].) So far, and since the arrows in Remark 1.5 are all strict inclusions, the class SES (and thus ES) has been shown to be 2 c -lineable.…”

Section: And For All Pairsmentioning

confidence: 96%

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Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps

Ciesielski

Gámez-Merino

Mazza

et al. 2016

Proc. Amer. Math. Soc.

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We investigate the additivity A A and lineability L \mathcal {L} cardinal coefficients for the following classes of functions: ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} of everywhere surjective functions that are not strongly everywhere surjective, Darboux-like, Sierpiński-Zygmund, surjective, and their corresponding intersections. The classes SES \operatorname {SES} and ES \operatorname {ES} have been shown to be 2 c 2^{\mathfrak {c}} -lineable. In contrast, although we prove here that ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} is c + {\mathfrak {c}}^+ -lineable, it is still unclear whether it can be proved in ZFC that ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} is 2 c 2^{\mathfrak {c}} -lineable. Moreover, we prove that if c \mathfrak {c} is a regular cardinal number, then A ( ES ∖ SES ) ≤ c A(\operatorname {ES} \setminus \operatorname {SES})\leq \mathfrak {c} . This shows that, for the class ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} , there is an unusually large gap between the numbers A A and L \mathcal {L} .

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Section: And For All Pairsmentioning

confidence: 96%

“…(ii) In [32] it is proved that CH implies that L(SZ ∩ AC) ≥ c ++ . A quick examination of the proof reveals that the argument also works under this weaker assumption and that it actually gives L(SZ ∩ AC ∩ ES) ≥ c ++ .…”

mentioning

confidence: 99%

Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps

Ciesielski

Gámez-Merino

Mazza

et al. 2016

Proc. Amer. Math. Soc.

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“…(Clearly, such a result cannot be proved in ZFC, as it is consistent that AC∩SZ = ∅.) It was noticed, in a 2017 paper [13] of Ciesielski, Gámez-Merino, Mazza, and Seoane-Sepúlveda (and repeated in a survey [18]), that the argument from [31] actually works under weaker assumption cov(M) = c. This, however, was not precise, since (as we will see below) the argument requires also the assumption that c is regular, which does not follow from cov(M) = c.…”

Section: Lemma 23mentioning

confidence: 99%

“…In 2015, Płotka, assuming CH, proved that the family AC∩SZ is c + -lineable [31]. (Clearly, such a result cannot be proved in ZFC, as it is consistent that AC∩SZ = ∅.)…”

Section: Lemma 23mentioning

confidence: 99%

Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

Ciesielski

Natkaniec

Rodríguez-Vidanes

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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A function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R is: almost continuous in the sense of Stallings, $$f\in \textrm{AC}$$ f ∈ AC , if each open set $$G\subset \mathbb {R}^2$$ G ⊂ R 2 containing the graph of f contains also the graph of a continuous function $$g:\mathbb {R}\rightarrow \mathbb {R}$$ g : R → R ; Sierpiński–Zygmund, $$f\in \textrm{SZ}$$ f ∈ SZ (or, more generally, $$f\in \textrm{SZ}(\textrm{Bor})$$ f ∈ SZ ( Bor ) ), provided its restriction $$f\restriction M$$ f ↾ M is discontinuous (not Borel, respectively) for any $$M\subset \mathbb {R}$$ M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R (i.e., an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC ) cannot be constructed in ZFC; however, an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC exists under the additional set-theoretical assumption $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c , that is, that $$\mathbb {R}$$ R cannot be covered by less than $$\mathfrak {c}$$ c -many meager sets. The primary purpose of this paper is to show that the existence of an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC is also consistent with ZFC plus the negation of $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c . More precisely, we show that it is consistent with ZFC+$${{\,\textrm{cov}\,}}(\mathcal {M})<\mathfrak {c}$$ cov ( M ) < c (follows from the assumption that $${{\,\textrm{non}\,}}(\mathcal {N})<{{\,\textrm{cov}\,}}(\mathcal {N})=\mathfrak {c}$$ non ( N ) < cov ( N ) = c ) that there is an $$f\in \textrm{SZ}(\textrm{Bor})\cap \textrm{AC}$$ f ∈ SZ ( Bor ) ∩ AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c or $${{\,\textrm{non}\,}}(\mathcal {N})<{{\,\textrm{cov}\,}}(\mathcal {N})=\mathfrak {c}$$ non ( N ) < cov ( N ) = c , the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.

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A century of Sierpiński–Zygmund functions

Ciesielski

Seoane‐Sepúlveda

2019

RACSAM

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Algebraic structures within subsets of Hamel and Sierpiński-Zygmund functions

Cited by 4 publications

References 3 publications

Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps

Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps

Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

A century of Sierpiński–Zygmund functions

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