Not all realcompact spaces are ultrapure

A general theorem (Theorem 1) concerning when spaces are not ultrapure in the sense of Arhangelʹskii is proved. Arhangelʹskii, in conversation, asked whether realcompactness implies his concept of ultrapurity and whether there are ZFC examples of astral spaces which are not ultrapure. Todorčević in (1984, Theorem 0.6) describes a class of spaces all of whose members are hereditarily realcompact. These spaces satisfy the hypothesis of Theorem 1 and are thus not ultrapure. Since some of these spaces are ZFC examples this answers both questions. These spaces and Theorem 1 are also applied, using an idea of Sakai (1986), to produce ZFC examples of spaces which are neat in Sakaiʹs sense but not pure in the sense of Arhangelʹskii. Sakai (1986) has a CH example of such a space.