More on perfectly normal non-realcompact spaces

We use the space ( ω 1 , τ ( C → ) ) associated with a guessing sequence C → on ω 1 to show that it is consistent with CH that there exists a locally countable, first-countable, locally compact, perfectly normal, non-realcompact space of size ℵ 1 which does not contain any sub-Ostaszewski spaces. By a similar technique, it is shown to be consistent with MA + ¬ CH that there exists a locally countable, first-countable, perfectly normal, non-realcompact space of size ℵ 1 .