Let \(\kappa\) be an infinite cardinal number. For any almost disjoint family \(\mathcal{A}\subset [\kappa]^{\omega}\) let \(\psi(\kappa,\mathcal{A})\) denote the space \(\kappa\cup\mathcal{A}\) with the topology having as a base all singletons \(\{\alpha\}\) for \(\alpha \mathfrak c\), and every \(\mathcal M \subset [\kappa]^{\omega}\), it is always the case that \(|\beta \psi (\kappa, \mathcal M) \setminus \psi (\kappa, \mathcal M)| \neq 1\), yet there exists a special free \(z\)-ultrafilter \(p\) on \(\psi (\kappa, \mathcal M)\) retaining some of the properties of \(p_{0}\). In particular both \(p\) and \(p_{0}\) have a clopen local base in \(\beta \psi \) (although \(\beta \psi (\kappa, \mathcal M)\) need not be zero-dimensional). A result for \(k > \mathfrak c\), that does not apply to \(p_{0}\), is that for certain \(\kappa > \mathfrak c, p\) is a P-point in \(\beta \psi \).''