Intersection properties of open sets (Q1068373)

Call a \(T_ 2\)-space \(P_ 1\) \((P_ 2)\) if whenever U and V are disjoint and open both (one of) U and V must be countable. The authors investigate the possible sizes of such spaces. They obtain the following results: (i) every \(P_ 2\)-space is of size at most \(2^{\omega}\), (ii) there is a \(P_ 1\)-space of size \(2^{\omega}\), (iii) there is a regular \(P_ 2\)-space of size \(2^{\omega}\), (iv) there is a compact \(P_ 2\)-space of size \(\omega_ 1\) and (v) if \(2^{\omega}\) is big then compact \(P_ 2\)-spaces of size \(2^{\omega}\) may or may not exist.