Cozero-accessible points (Q1032899)

According to the author a point \(x\) in a space \(X\) is said to be cozero-accessible if for each dense open set \(U\subset X\) there is a cozero-set \(C\subset U\) such that \(x \in \overline{C}\). It is consistent with \(MA + \neg CH\) that there are cozero-accessible points in \(\beta \mathbb N \setminus \mathbb N\), while \(PFA\) implies that there are no such points in \(\beta \mathbb N \setminus \mathbb N\). Five interesting questions are posed.