Fin-intersecting MAD families and the pseudocompactness of hyperspaces
We study Ginsburg's questions on the relations between the pseudocompactness of the Vietoris hyperspace of a given topological space and the pseudocompactness of its powers restricted to the context of Isbell-Mrówka space. An almost disjoint family is said to be pseudocompact if the Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact. In the context of Isbell-Mrówka spaces, Ginsburg's questions become questions about the existence of pseudocompact MAD families. We will discuss what has been done around this problem and what remains open.
To further study this problem we propose a new class of almost disjoint families which we call fin-intersecting almost disjoint families. Every fin intersecting MAD family is pseudocompact. Under p=c such families exist generically, but there is also a MAD pseudocompact family which is not fin-intersecting. Also, under CH, forcing extensions obtained by adding Cohen reals and Random reals contain such families.