Fin-intersecting MAD families

Cesar Corral, Vinicius Rodrigues


We introduce a new class of almost disjoint families which we call fin-intersecting almost disjoint families. They are related to almost disjoint families whose Vietoris Hyperspace of their Isbell-Mrówka spaces are pseudocompact. We show that under $\mathfrak p=\mathfrak c$ fin-intersecting MAD families exist generically and they also exist if $\mathfrak{a<s}$, but that there are also non fin-intersecting MAD families in ZFC. We also show that under CH, there exist fin intersecting MAD families which remain like that after adding an arbitrary quantity of Cohen reals and Random reals. These results give more models in which pseudocompact MAD families exist.