Some remarks on second category sets.

*(English)*Zbl 0827.03031We give some loosely connected consistency results concerning second category sets. In “Large small sets” [Colloq. Math. 56, No. 2, 231-233 (1988; Zbl 0687.03033)], the author proved, extending some earlier results of Mycielski, that \(\text{MA}_\kappa\) implies that there exists a measure zero, first category set \(X\) such that if \(Y\) is a set of reals of size \(\leq \kappa\) then \(Y \subseteq X+ c\) for some real \(c\). K. Muthuvel [Real Anal. Exch. 17, No. 2, 771-774 (1992; Zbl 0761.28001)] gave some applications of this construction and asked if it follows already from ZFC that such an \(X\) exists for every measure zero, first category set \(Y\) of size \(<2^\omega\). We show that this is not the case. Namely, it is consistent that \(2^\omega= \omega_2\) and for no first category \(X\) it is true that for every first category, measure zero set \(Y\) of size \(\omega_1\) we have \(Y \subseteq X+c\) for some \(c\).

U. Avraham [Isr. J. Math. 39, 167-176 (1981; Zbl 0489.03019)] proved that it is consistent that \(2^\omega= \omega_2\) and there is a set mapping \(f: \mathbb{R}\to P(\mathbb{R})\) such that \(f(x)\) is nowhere dense for \(x\in \mathbb{R}\), and there is no uncountable free set for \(f\). Also, \(\text{MA}_{\omega_1}\) is consistent with the statement that every set mapping as above has an uncountable free set. For set mappings where the images are first category sets, the situation is not as easy. If \(2^\omega= \omega_2\), an easy well-ordering argument gives that there is a set mapping \(f: \mathbb{R}\to [\mathbb{R} ]^{\omega_1}\) with no free sets of size 2. If \(\text{MA}_{\omega_1}\) holds, the images are of first category. Here we prove that it is consistent that \(\text{MA}_{\omega_1}\) holds, and there is a set \(X\) of size \(2^\omega= \omega_3\) such that if \(Y\subseteq X\) is of first category, then \(|Y|\leq \omega_1\). This implies, by a result of Ruziewicz, that if \(f\) is a set mapping on \(\mathbb{R}\) with first category images, then there is a (second category) free set of size \(\aleph_3\).

The last theorem we address is the following. Is it true that there is an almost disjoint family of second category sets which is of cardinality \(>2^\omega\)? Here, almost disjoint means that the intersection of any two members in the family is of first category. W. Sierpiński [Hypothèse du continu (Monogr. Mat. 4, Warszawa) (1934; Zbl 0009.30201)] proved that if CH holds, then there are \(\aleph_2\) second category sets with pairwise countable intersection. It is easy to show that there exist \(2^\omega\) disjoint second category sets. We show that it is consistent that \(2^\omega= \omega_2\) and there are no \(\aleph_3\) almost disjoint second category sets.

U. Avraham [Isr. J. Math. 39, 167-176 (1981; Zbl 0489.03019)] proved that it is consistent that \(2^\omega= \omega_2\) and there is a set mapping \(f: \mathbb{R}\to P(\mathbb{R})\) such that \(f(x)\) is nowhere dense for \(x\in \mathbb{R}\), and there is no uncountable free set for \(f\). Also, \(\text{MA}_{\omega_1}\) is consistent with the statement that every set mapping as above has an uncountable free set. For set mappings where the images are first category sets, the situation is not as easy. If \(2^\omega= \omega_2\), an easy well-ordering argument gives that there is a set mapping \(f: \mathbb{R}\to [\mathbb{R} ]^{\omega_1}\) with no free sets of size 2. If \(\text{MA}_{\omega_1}\) holds, the images are of first category. Here we prove that it is consistent that \(\text{MA}_{\omega_1}\) holds, and there is a set \(X\) of size \(2^\omega= \omega_3\) such that if \(Y\subseteq X\) is of first category, then \(|Y|\leq \omega_1\). This implies, by a result of Ruziewicz, that if \(f\) is a set mapping on \(\mathbb{R}\) with first category images, then there is a (second category) free set of size \(\aleph_3\).

The last theorem we address is the following. Is it true that there is an almost disjoint family of second category sets which is of cardinality \(>2^\omega\)? Here, almost disjoint means that the intersection of any two members in the family is of first category. W. Sierpiński [Hypothèse du continu (Monogr. Mat. 4, Warszawa) (1934; Zbl 0009.30201)] proved that if CH holds, then there are \(\aleph_2\) second category sets with pairwise countable intersection. It is easy to show that there exist \(2^\omega\) disjoint second category sets. We show that it is consistent that \(2^\omega= \omega_2\) and there are no \(\aleph_3\) almost disjoint second category sets.

##### MSC:

03E35 | Consistency and independence results |

28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |