Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions



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Krzysztof, Chris and Natkaniec, T. and Rodríguez-Vidanes, D.L. Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions. (Unpublished)

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A function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC, if each open set G ⊂ R2 containing the graph of f contains also the graph of a continuous function g : R → R; Sierpiński-Zygmund, f ∈ SZ (or, more generally, f ∈ SZ(Bor)), provided its restriction f M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński-Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov(M) = c, that is, that R cannot be covered by less than c-many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ∩AC is also consistent with ZFC plus the negation of cov(M) = c. More precisely, we show that it is consistent with ZFC+cov(M) < c (follows from the assumption that non(N ) < cov(N ) = c) that there is an 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 cov(M) = c or 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.

Item Type:Article
Uncontrolled Keywords:Additivity Almost continuous functions; Covering of category; Covering of measure; Lineability; Random reals; Sierpiński-Zygmund functions
Palabras clave (otros idiomas):Espacios vectoriales
Subjects:Sciences > Mathematics
Sciences > Mathematics > Mathematical analysis
ID Code:73501
Deposited On:07 Jul 2022 11:28
Last Modified:07 Jul 2022 12:36

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