OPEN PROBLEMS IN TOPOLOGY

Here are some problems that interest me. Most I have worked on; some Ihave not. I have in general avoided listing well-known problems that I amnot particularly associated with, since they surely will be covered elsewherein this volume.A. Normal Moore Space ProblemsTall [1984, 1979] and Fleissner [1984] are good references for normal Moorespaces.Question A1. Is it consistent with 2 ℵ0 = ℵ 2 that every normal Moore space 36. ?is metrizable?It is known to be consistent with 2 ℵ0 being weakly inaccessible (Nyikos[1982], Dow, Tall and Weiss [19∞b]). If—as once conjectured by SteveWatson—2 ℵ0 not weakly inacessible implies the existence of a normal nonmetrizableMoore space, there would be a simple proof of the necessity oflarge cardinals to prove the consistency of the Normal Moore Space Conjecture.The game plan would be to work with Fleissner’s CH example of anormal non-metrizable Moore space (Fleissner [1982]) and weaken the hypothesis.However, Fleissner and I conjecture the other way—namely that theConjecture is consistent with 2 ℵ0 = ℵ 2 . In particular I conjecture that theConjecture holds in the model obtained by Mitchell-collapsing a supercompactcardinal. (For Mitchell collapse, see Mitchell [1972] and Abraham [1983].)There are enough Cohen reals in this model so that normal Moore spacesof cardinality ℵ 1 are metrizable (Dow, Tall and Weiss [19∞b]), so thisconjecture is a “reflection problem”—see below.Question A2. Is it consistent with GCH that normal Moore spaces are 37. ?para-Lindelöf?Aspaceispara-Lindelöf if every open cover has a locally countable openrefinement. This is an attempt to get as much of the Normal Moore SpaceConjecture as possible consistent with GCH. It is done for spaces of cardinality≤ℵ 1 in Tall [1988]. Any consistency proof would likely establishthe consistency with GCH of every first countable countably paracompactsubmetacompact space being para-Lindelöf. Again, this is done for spacesof cardinality ≤ ℵ 1 in Tall [1988]; indeed, first countability is weakenedto character ≤ℵ 1 . It’s consistent with GCH that there’s a normal Moorespace that’s not collectionwise Hausdorff, hence not para-Lindelöf (Devlinand Shelah [1979]).Question A3. Does the existence of a normal non-metrizable Moore space 38. ?imply the existence of one which is in addition is metacompact?23