Cardinal coefficients related to surjectivity, Darboux, and Sierpiński-Zygmund maps

Abstract:

We investigate the additivity $A$ and lineability $\mathcal {L}$ cardinal coefficients for the following classes of functions: $\operatorname {ES} \setminus \operatorname {SES}$ of everywhere surjective functions that are not strongly everywhere surjective, Darboux-like, Sierpiński-Zygmund, surjective, and their corresponding intersections. The classes $\operatorname {SES}$ and $\operatorname {ES}$ have been shown to be $2^{\mathfrak {c}}$-lineable. In contrast, although we prove here that $\operatorname {ES} \setminus \operatorname {SES}$ is ${\mathfrak {c}}^+$-lineable, it is still unclear whether it can be proved in ZFC that $\operatorname {ES} \setminus \operatorname {SES}$ is $2^{\mathfrak {c}}$-lineable. Moreover, we prove that if $\mathfrak {c}$ is a regular cardinal number, then $A(\operatorname {ES} \setminus \operatorname {SES})\leq \mathfrak {c}$. This shows that, for the class $\operatorname {ES} \setminus \operatorname {SES}$, there is an unusually large gap between the numbers $A$ and $\mathcal {L}$.