On a problem concerning generalizations of realcompact spaces

PF-compact spaces are defined. Every almost realcompact space is PF-compact and every separable PF-compact space is almost realcompact. Nonetheless in the constructible universe V = L, a space of cardinality ?₁ is PF-compact if, and only if, the closure of each of its countable subsets is almost realcompact, and ω₁ is hereditarily PF-compact.