Dense normality and Sorgenfrei n-space
by
Brian McMaster
Belfast
Coauthors: Aisling McCluskey (Galway)

A space is kappa-normal if each two disjoint closed sets that are 'canonical', that is, that are the closures of open sets, possess disjoint neighbourhoods. A set A is concentrated on a set D if their intersection is dense in A. A space is densely normal if it has a dense subset D such that each two disjoint closed sets that are concentrated on D possess disjoint neighbourhoods.

Clearly, densely normal implies kappa-normal. In response to a question of Arhangel'skii, Just and Tartir constructed a regular space that is kappa-normal but not densely normal. McCluskey showed [in ZFC] that the Sorgenfrei plane [the product of two copies of the Sorgenfrei line] is a much simpler example of this phenomenon.

We extend the range of 'easy counterexamples' here by showing that, for finite n > 2, Sorgenfrei n-space is not densely normal.

Date received: August 19, 2002