Semi-stratifiable spaces with monotonically normal compactifications

Abstract

In this paper we use Mary Ellen Rudin's solution of Nikiel's problem to investigate metrizability of certain subsets of compact monotonically normal spaces. We prove that if H is a semi-stratifiable space that can be covered by a σ-locally-finite collection of closed metrizable subspaces and if H embeds in a monotonically normal compact space, then H is metrizable. It follows that if H is a semi-stratifiable space with a monotonically normal compactification, then H is metrizable if it satisfies any one of the following: H has a σ-locally finite cover by compact subsets; H is a σ-discrete space; H is a scattered; H is σ-compact. In addition, a countable space X has a monotonically normal compactification if and only if X is metrizable. We also prove that any semi-stratifiable space with a monotonically normal compactification is first-countable and is the union of a family of dense metrizable subspaces. Having a monotonically normal compactification is a crucial hypothesis in these results because R.W. Heath has given an example of a countable non-metrizable stratifiable (and hence monotonically normal) group. We ask whether a first-countable semi-stratifiable space must be metrizable if it has a monotonically normal compactification. This is equivalent to “If X is a first-countable stratifiable space with a monotonically normal compactification, must H be metrizable?”