\(P\)-points in \(\mathbb{N}^*\) and the spaces of continuous functions (Q1295305)

Let \(\mathbb{N}_{\phi}\) denote the subspace \(\mathbb{N}\cup\{\phi\}\) of the Čech-Stone compactification \(\beta \mathbb{N}\), where \(\phi\in\mathbb{N}^*=\beta \mathbb{N}\smallsetminus\mathbb{N}\). It is known that \(C_p(\mathbb{N}_{\phi})\) is a Baire space for every \(\phi\), where \(C_p(\cdot)\) is the space of continuous functions with pointwise topology. The authors prove that \(C_p(\mathbb{N}_{\phi})\) is a hereditary Baire space if and only if \(\phi\) is a \(P\)-point in \(\mathbb{N}^*\). This property is also shown to be equivalent to the space \(\mathbb{Q}\) of rational numbers being not embeddable in \(C_p(\mathbb{N}_{\phi})\) as a closed subspace. In addition, characterizations of \(\phi\) being a \(P\)-point are given in terms of games played in \(\mathbb{N}_{\phi}\) and in \(C_p(\mathbb{N}_{\phi})\).