Abstract
Assuming that continuum \(\mathfrak {c}\) is a regular cardinal, we show that the class \({{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}\) of all functions from \(\mathbb {R}\) to \(\mathbb {R}\) that are perfectly everywhere surjective (so Darboux) but not connectivity is \(\mathfrak {c}^+\)-lineable, that is, that there exists a linear space of \(\mathbb {R}^\mathbb {R}\) of cardinality \(\mathfrak {c}^+\) that is contained in \(({{\,\mathrm{PES}\,}}{\setminus }{{\,\mathrm{Conn}\,}})\cup \{0\}\). Moreover, assuming additionally that \(\mathbb {R}\) is not a union of less than \(\mathfrak {c}\)-many meager sets, we prove \(\mathfrak {c}^+\)-lineability of the class \({{\,\mathrm{SZ}\,}}\cap {{\,\mathrm{ES}\,}}{\setminus }{{\,\mathrm{Conn}\,}}\) of Sierpiński-Zygmund everywhere surjective but not connectivity functions.
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Notes
Nash shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he also shared the Abel Prize with Louis Nirenberg for his work on nonlinear PDEs.
Recall that \(\mathfrak {c}\) is regular when the union of less than \(\mathfrak {c}\)-many sets, each of cardinality less than \(\mathfrak {c}\), has cardinality less than \(\mathfrak {c}\).
This follows from the fact that \( {{\,\mathrm{SZ}\,}}({{\,\mathrm{\mathcal {B}}\,}})={{\,\mathrm{SZ}\,}}(\{f\restriction X:f\in {{\,\mathrm{\mathcal {B}}\,}}\ \& \ X\subset \mathbb {R}\})\) justified by a theorem of Kuratowski (see e.g. [26, p. 73]) that every partial Borel map from \(X\subset \mathbb {R}\) into \(\mathbb {R}\) can be extended to a Borel function from \(\mathbb {R}\) to \(\mathbb {R}\).
Actually, every infinite dimensional vector space V has such subset M: if B is an infinite linearly independent subset of V, sets \(\{B_n\subset B:n<\omega \}\) are pairwise disjoint with each \(B_n\) having n elements, and each \(V_n\) is spanned by \(B_n\), then \(M=\bigcup _{n<\omega } V_n\) is as needed.
\(A(\mathcal {F}):=\min \bigl (\{|F|:F\subset \mathbb {R}^\mathbb {R} \text{ and } \varphi +F\not \subset \mathcal {F} \text{ for } \text{ every } \varphi \in \mathbb {R}^\mathbb {R}\}\cup \{(2^{\mathfrak {c}})^+\}\bigr )\), where \(\mathcal {F}\subset \mathbb {R}^\mathbb {R}\).
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Albkwre, G.M., Ciesielski, K.C. & Wojciechowski, J. Lineability of the functions that are Sierpiński–Zygmund, Darboux, but not connectivity. RACSAM 114, 145 (2020). https://doi.org/10.1007/s13398-020-00881-9
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DOI: https://doi.org/10.1007/s13398-020-00881-9