Conditional completeness of \(C(X,\mathbb{R}_\tau)\) for weak \(P\)-spaces \(\mathbb{R}_\tau\) (Q1029975)

A topological space \(X\) is called a \(P\)-space if the set \(Z(f) = f^{-1}(0)\) is open for all \(f\in C(X)\) (\(C(X)\) denotes the family of all continuous real functions on \(X\)). A space \(X\) is called a weak \(P\)-space if each countable subset is closed. Let \(\tau \) be a topology on \(\mathbb R\) which is finer than the usual topology. The author compares conditional completeness and conditional \(\sigma \)-completeness of \(C(X)\) and \(C(X,\mathbb R_\tau)\). If \(X\) is a zero-dimensional space then \(C(X,\mathbb R_\tau)\) is conditionally \(\sigma\)-complete if and only if \(X\) is a \(P\)-space and \(C(X,\mathbb R_\tau)\) is conditionally complete if and only if \(X\) is an extremally disconnected \(P\)-space.