$P$-sets and minimal right ideals in $\mathbb {N}^{*}$
Volume 229 / 2015
Fundamenta Mathematicae 229 (2015), 277-293
MSC: Primary 54D35, 03E35; Secondary 03E17, 37B99, 06A07, 22A15.
DOI: 10.4064/fm229-3-4
Abstract
Recall that a $P$-set is a closed set $X$ such that the intersection of countably many neighborhoods of $X$ is again a neighborhood of $X$. We show that if $\mathfrak {t}= \mathfrak {c}$ then there is a minimal right ideal of $(\beta \mathbb {N},+)$ that is also a $P$-set. We also show that the existence of such $P$-sets implies the existence of $P$-points; in particular, it is consistent with ZFC that no minimal right ideal is a $P$-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift map and its inverse are (up to isomorphism) the unique chain transitive autohomeomorphisms of $\mathbb {N}^*$.
