Tomasz Natkaniec: Perfectly everywhere surjective but not Jones functions

Tuesday, November 14, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Tomasz Natkaniec (University of Gdansk)

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.

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