Fernando Hernández-Hernández

arXiv2023

Generalized independence

We explore different generalizations of the classical concept of independent families on $ω$ following the study initiated by Fisher and Montoya. We show that under $\diamondsuit^*(κ)$ we can get strongly independent families on $κ$ of size $2^κ$ and present an equivalence of $\mathsf{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $\mathcal{C}$-independent families, where $\mathcal{C}$ is the club filter. Also we show a relationship between the existence of $\mathcal{J}$-independent families and the saturation of the ideal $\mathcal{J}$.

math.LO↗

arXiv2018

$\mathfrak{c}$-many types of a $Ψ$-space

We show that for any cardinal $ω<κ\leq \mathfrak{c}$ with $cf(κ) > ω$, there are $\mathfrak{c}$ many AD families whose $Ψ$-spaces are pairwise non-homeomorphic and they can be Luzin families or branch families of $2^ω$.

math.GN↗