Nonmeasurable Cardinals and Pointfree Topology

Abstract This paper establishes that the familiar rôle of nonmeasurable cardinals in classical topology extends to pointfree topology, that is, the setting of frames. For this, it considers the frames which are the pointfree form of the extremally disconnected P-spaces, namely the extremally disconnected 0-dimensional frames in which any countable join of complemented elements is complemented, and shows that they (1) have discrete spectrum and (2) are realcompact whenever they have nonmeasurable cardinal. An important tool obtained for this purpose is the result that, for a Boolean frame L, any σ-frame homomorphism L → 2 preserves the joins of all subsets of nonmeasurable cardinal.