An application of Mary Ellen Rudin's solution to Nikiel's Conjecture

Abstract

The concept of utter normality is a formal strengthening of the concept of monotone normality. It is an unsolved problem whether every monotonically normal space is utterly normal. Here we show that every suborderable monotonically normal space and every compact monotonically normal space is utterly normal. The latter result makes use of Mary Ellen Rudin's powerful theorem that every compact, monotonically normal space is the continuous image of a compact linearly ordered space. Corollaries include the fact that a compact, monotonically normal space is hereditarily weakly orthocompact. We show that a locally compact, monotonically normal space has a monotonically normal compactification if, and only if, it is weakly orthocompact. Mary Ellen Rudin's example [13] of a locally compact, monotonically normal space whose one point compactification is not monotonically normal shows that some such characterization is necessary.