A MAD Q-set

Arnold W. Miller¹

¹ Department of Mathematics, Van Vleck Hall University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53706-1388, U.S.A.

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 271-281

Résumé

A MAD (maximal almost disjoint) family is an infinite subset ${\mathcal A}$ of the infinite subsets of $\omega =\{0,1,2,\ldots\}$ such that any two elements of ${\mathcal A}$ intersect in a finite set and every infinite subset of $\omega $ meets some element of ${\mathcal A}$ in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative $G_\delta $-set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of $P(\omega )=2^{\omega }$.

Détail

DOI : 10.4064/fm178-3-6

Keywords: mad maximal almost disjoint family infinite subset mathcal infinite subsets omega ldots elements mathcal intersect finite set every infinite subset omega meets element mathcal infinite set q set uncountable set reals every subset relative delta set shown relatively consistent zfc there exists mad family which q set topology inherits subset omega omega