$P$-sets and minimal right ideals in $\mathbb {N}^{*}$

W. R. Brian¹

¹ Department of Mathematics Tulane University 6823 St. Charles Ave. New Orleans, LA 70118, U.S.A.

Fundamenta Mathematicae, Tome 229 (2015) no. 3, pp. 277-293

Résumé

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 $\mathbb {N}^*$.

Détail

DOI : 10.4064/fm229-3-4

Keywords: recall p set closed set intersection countably many neighborhoods again neighborhood mathfrak mathfrak there minimal right ideal beta mathbb p set existence p sets implies existence p points particular consistent zfc minimal right ideal p set application these results prove consistent independent zfc shift map its inverse isomorphism unique chain transitive autohomeomorphisms nbsp mathbb *