Some more \(\kappa\)-fully normal spaces. (English) Zbl 0636.54022

Hart, Klaas Pieter

Rend. Circ. Mat. Palermo, II. Ser. Suppl. 11, 55-59 (1985).

The author’s introduction: “In Trans. Am. Math. Soc. 86, 489-505 (1957; Zbl 0078.148) M. J. Mansfield introduced the notions of \(\kappa\)- full and almost \(\kappa\)-full normality. One of the problems still left unanswered is the solution of the equation: \[ \kappa \quad -\quad \text{full normality = almost-\(\kappa\)} \quad -\quad \text{full normality + \({\mathcal P}\).} \] In “Covering properties and quasi-uniformities of topological spaces”, Thesis VPI, 1978, Junnila essentially showed that almost-\(\kappa\)-fully normal orthocompact spaces are \(\kappa\)-fully normal, so the problem was raised whether orthocompactness might be a solution of the above equation (it was the only known candidate). The purpose of this note is to show that this is, at least consistently, not the case. We find under GCH (actually somewhat weaker assumptions suffice) that for every \(\kappa\) there is a \(\kappa\)-fully normal space which is not orthocompact.”

Reviewer: C.E.Aull

MSC:

54D20

Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

almost \(\kappa \)-full normality; orthocompactness; GCH; \(\kappa \)-fully normal space

Citations:

Zbl 0078.148