Selectively \((a)\)-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle (Q2834139)

The paper studies the following two basic notions.NEWLINENEWLINEA topological space \(X\) has property \((a)\) iff for every open cover \(U\) and every dense subset \(D\) of \(X\) there is a closed and discrete \(A\) of \(D\) such that \(X = \bigcup \{O \in U_n: O \cap A_n \neq \emptyset\}\).NEWLINENEWLINEA topological space \(X\) is a selectively \((a)\) space iff for every sequence \(U_n\) of open covers and every dense subset \(D\) of \(X\) there is a sequence \(A_n\) of closed and discrete subsets of \(D\) such that the open sets \(O_n = \bigcup \{O \in U_n: O \cap A_n \neq \emptyset\}\) form an open cover of \(X\).NEWLINENEWLINEPrior work of the second author [Acta Math. Hung. 142, No. 2, 420--432 (2014; Zbl 1299.54046)] had established that certain topological spaces \(X\) derived from almost disjoint families of subsets of the natural numbers satisfy the following: If \(X\) is a selectively \((a)\)-space and \(H\) is a closed and discrete subset of \(X\) then \(|H| < 2^{d(X)}\), where \(d(X)\) is the minimum cardinality of a dense subset of \(X\). This implies that the almost disjoint family from which \(X\) is constructed has a cardinality strictly below the continuum whenever \(X\) is a selectively \((a)\) space. The second author in [loc. cit.] then asked whether \(X\) under the hypothesis that \(X\) is a selectively \((a)\)-space implies that the family is countable iff the continuum hypothesis is true. In the present work, the authors show that this is not the case and that the fact that the almost disjoint family has to be countable is consistent with CH being false and they establish a sufficient condition which does not imply CH for this fact.