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

Abstract

We investigate the additivity A A and lineability L \mathcal {L} cardinal coefficients for the following classes of functions: ES ∖ SES \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 SES \operatorname {SES} and ES \operatorname {ES} have been shown to be 2 c 2^{\mathfrak {c}} -lineable. In contrast, although we prove here that ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} is c + {\mathfrak {c}}^+ -lineable, it is still unclear whether it can be proved in ZFC that ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} is 2 c 2^{\mathfrak {c}} -lineable. Moreover, we prove that if c \mathfrak {c} is a regular cardinal number, then A ( ES ∖ SES ) ≤ c A(\operatorname {ES} \setminus \operatorname {SES})\leq \mathfrak {c} . This shows that, for the class ES ∖ SES \operatorname {ES} \setminus \operatorname {SES} , there is an unusually large gap between the numbers A A and L \mathcal {L} .