OPEN PROBLEMS IN TOPOLOGY

1. IntroductionThis assortment is a list of problems that I worked on between 1979 and 1989which I failed to solve. Some of the problems are due to other topologistsand set theorists and I have attributed them when references are available inthe literature. An earlier list appeared in 1984 in the Italian journal, Rend.Circ. Mat. Palermo. This list was entitled “Sixty questions on regular nonparacompactspaces”. Thirteen of these questions have since been answered(we shall give the numbering in that earlier paper in each case).• Carlson, in [1984], used Nyikos’ solution from Nyikos [1980] of thenormal Moore space problem to show that if it is consistent that thereis a weakly compact cardinal then it is consistent that normal Moorespaces of cardinality at most 2 ℵ0 are metrizable. This solved Palermo #11.• In [19∞a] Balogh showed at the STACY conference at York Universitythat assuming the consistency of the existence of a supercompactcardinal, it is consistent that normal locally compact spaces are collectionwisenormal thus solving Palermo #16. Tall had earlier establishedthis result for spaces of cardinality less than ℶ ω . See Tall’s B1.• In [1985] Daniels and Gruenhage constructed a perfectly normallocally compact collectionwise Hausdorff space under ♦ ∗ which is notcollectionwise normal, thus answering Palermo #22, Palermo #23 andPalermo #24 in one blow.• Balogh showed that under V = L countably paracompact locally compactspaces are collectionwise Hausdorff and that under V = L countablyparacompact locally compact metacompact spaces are paracompactthus answering Palermo #26 and giving a partial solution to Palermo #28.• Burke showed that under PMEA, countably paracompact Moore spacesare metrizable thus solving a famous old problem and incidentally answeringPalermo #30, Palermo #33 and Palermo #34.• It turned out that Palermo #37 and Palermo #47 were somewhat illposedsince Fleissner’s CH space already in existence at that time answeredboth in its ZFC version by being a para-Lindelöf metacompactnormal space of character 2 ℵ0 which is not collectionwise normal.• Daniels solved Palermo #56 by showing that in ZFC the Pixley-Royspace of the co-countable topology on ω 1 is collectionwise Hausdorff (itwas her question to begin with).39