Maximal almost disjoint families and pseudocompactness of hyperspaces

2020/04/19 by Guzmán, Osvaldo, Hrušák, Michael, Rodrigues, Vinicius de Oliveira +2
#03E17 #03E35 #FOS: Mathematics #General Topology (math.GN) #Primary 54D20 #Secondary 54D35

paper · doi:10.48550/arxiv.2004.08760

Abstract

We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if MA_\mathfrak c (\mathcal P(ω)/fin) holds. We further study the question whether there is a maximal almost disjoint family whose hyperspace is pseudocompact and prove that consistently such families do not exist genericaly, by constructing a consistent example of a maximal almost disjoint family \mathcal A of size less than \mathfrak c whose hyperspace is not pseudocompact.

Related

• New examples of MAD families with pseudocompact hyperspaces
• Fin-intersecting MAD families
• Two more characterisations of nearly pseudocompact space
• Construction of nearly pseudocompactification
• Comparing the closed almost disjointness and dominating numbers