<p>This paper investigates the concept of lineability for families of real-valued functions, a notion first introduced by Vladimir Gurariǐ in the 1960s. Assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(|\mathbb {R}|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> is a regular cardinal and that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> cannot be covered by less than <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(|\mathbb {R}|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>-many meager sets. Our main results establish that the families of Sierpiński-Zygmund functions in three different Darboux-like classes are at least <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathfrak {c}^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>-lineable, yielding lineability coefficients strictly bounded below by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathfrak {c}^+\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="fraktur">c</mi> </mrow> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>. Since it is known that these families can be maximally or minimally lineable in different models, our findings also provide insight into the relationship between set-theoretical assumptions and the lineability of function spaces.</p>

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\(\mathfrak {c}^+\)-Lineability of Sierpiński-Zygmund functions that are Darboux but not connectivity

  • Gbrel Albkwre,
  • Cheng-Han Pan

摘要

This paper investigates the concept of lineability for families of real-valued functions, a notion first introduced by Vladimir Gurariǐ in the 1960s. Assume that \(|\mathbb {R}|\) | R | is a regular cardinal and that \(\mathbb {R}\) R cannot be covered by less than \(|\mathbb {R}|\) | R | -many meager sets. Our main results establish that the families of Sierpiński-Zygmund functions in three different Darboux-like classes are at least \(\mathfrak {c}^+\) c + -lineable, yielding lineability coefficients strictly bounded below by \(\mathfrak {c}^+\) c + . Since it is known that these families can be maximally or minimally lineable in different models, our findings also provide insight into the relationship between set-theoretical assumptions and the lineability of function spaces.