Primitive shifts on \(\psi \)-spaces (Q409670)

The sets \(A\) and \(B\) are called almost disjoint if \(A\cap B=\) a finite set. Let \(D\) be a countable set with the discrete topology. A family \({\mathcal F}\) of infinite subsets of \(D\) is called almost disjoint if every pair of elements of \({\mathcal F}\) is almost disjoint. The almost disjoint family \({\mathcal F}\) is called a maximal almost disjoint family if it is maximal among such families. MAD is the abbreviation for ``maximal almost disjoint''. If \({\mathcal F}\) is a MAD family, then \(D\cup{\mathcal F}\) is called a \(\Psi\)-space and its one-point compactification is called a \(\Psi^*\)-space. The family \({\mathcal F}\) is said to be \(\sigma\)-invariant if for every \(A\in{\mathcal F}\), the sets \(\sigma(A)\) and \(\sigma^{-1}(A)\) are elements of \({\mathcal F}\). The authors establishNEWLINENEWLINE Theorem 1. Let \(\sigma\) be a primitive shift on a countable discrete space \(D\). There are \(2^c\) maximal almost disjoint families of infinite subsets of \(D\) that are \(\sigma\)-invariant.NEWLINENEWLINE Theorem 2. There are \(2^c\) nonhomeomorphic \(\Psi^*\)-spaces that admit a primitive shift.NEWLINENEWLINE Theorem 3. If the almost disjointness number is \(c\), then there exists a \(\Psi\)-space with no primitive shifts.