An equivalent condition for the existence of a measurable cardinal.

Abstract

Introduction. The main purpose of this paper is to prove the equivalence of the existence of a topological space in which there is a nonconvergent open ultrafilter with the countable intersection property and the existence of a measurable cardinal. This paper will answer indirectly the open question in [2 ]. 1.1 DEFINITION. We call a cardinal m measurable if a set X of cardinal m admits a { 0, 1 } valued measure ,u such that ,u(X) =1 and ,u({x }) = 0 for every xEX. For any set X, I XI will denote the cardinal of X. 1.2 DEFINITION. An open filter is a nonempty collection of open sets cU such that (1) 0 El, and (2) if U, VEza and G=int(G) D UnV, then G Ez A. An open ultrafilter is an open filter which is maximal in the collection of open filters.