Tomasz Natkaniec

Tomasz Natkaniec: Perfectly everywhere surjective but not Jones functions

Tuesday, November 14, 2017 17:15

Room: D1-215

Tomasz Natkaniec

Title: Perfectly everywhere surjective but not Jones functions

Abstract. Given a function \(f:\mathbb{R}\to\mathbb{R}\) we say that

  1. \(f\) is perfectly surjective (\(f\in \mathrm{PES}\)) if \(f[P]=\mathbb{R}\) for every perfect set \(P\);

  2. \(f\) is a Jones function (\(f\in\mathrm{J}\)) if \(C\cap f\neq\emptyset\) for every closed \(C\subset\mathbb{R}^2\) with \(\mathrm{dom}(C)\) of size \(\mathfrak{c}\).

M. Fenoy-Munoz, J.L. Gamez-Merino, G.A. Munoz-Fernandez and E. Saez-Maestro in the paper A hierarchy in the family of real surjective functions [Open Math. 15 (2017), 486--501] asked about the lineability of the set \(\mathrm{PES}\setminus\mathrm{J}\).
Answering this question we show that the class \(\mathrm{PES}\setminus\mathrm{J}\) is \(\mathfrak{c}^+\)-lineable. Moreover, if
\(2^{<\mathfrak{c}}=\mathfrak{c}\) then \(\mathrm{PES}\setminus\mathrm{J}\) is \(2^\mathfrak{c}\)-lineable. We prove also that the additivity number
\(A(\mathrm{PES}\setminus\mathrm{J})\) is between \(\omega_1\) and \(\mathfrak{c}\). Thus \(A(\mathrm{PES}\setminus\mathrm{J})=\mathfrak{c}\) under CH,
however this equality can't be proved in ZFC, because the Covering Property Axiom CPA implies \(A(\mathrm{PES}\setminus\mathrm{J})=\omega_1<\mathfrak{c}\).

The talk is based on the joint paper:
K.C.Ciesielski, J.L. Gamez-Merino, T. Natkaniec, and J.B.Seoane-Sepulveda, On functions that are almost continuous and perfectly everywhere surjective but not Jones. Lineability and additivity, submitted.