Maximal almost disjoint families and pseudocompactness of hyperspaces

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

We study the relations between a generalization of pseudocompactness, named $(κ, M)$-pseudocompactness, the countably compactness of subspaces of $βω$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $βω$ that is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $ω\subseteq X\subseteq βω$ is such that $X$ is $(\mathfrak c, ω^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by …

Small MAD families whose Isbell-Mrówka spaces are pseudocompact

Given a countable transitive model $M$ for ZFC+CH, we prove that one can produce a maximal almost disjoint family in $M$ whose Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact on every Cohen extension of $M$. We also show that a classical example of $ω_1$-sized maximal almost disjoint family obtained by a forcing iteration of length $ω_1$ in a model of non CH is such that the Vietoris Hyperspace of its Isbell-Mrówka space is pseudocompact.

Fin-相交 MAD 族

Fin-intersecting MAD families

本文定义 fin-相交几乎不交族，刻画其与 Isbell-Mrówka 空间 Vietoris 超空间伪紧性的关系，并在 p=c、a<s、CH 加力等情形证明其存在性。