Remote Points in βR

Abstract

The proof turns out to be considerably more difficult than anticipated. If we assume the continuum hypothesis (designated [CH]), then we can find such a point p (2.5); however, we do not know whether the continuum hypothesis is necessary. The result is obtained as a consequence of a more general theorem: for a suitably restricted class of spaces X, a point with a somewhat stronger property exists if and only if X admits an unbounded continuous function (2.3). A byproduct is: [CH] there exists a countable, completely regular space without isolated points, one of whose points is not a limit point of any discrete set (2.6).