OPEN PROBLEMS IN TOPOLOGY

§F] Reflection Problems 29establishes consistency for spaces that in addition are locally of cardinality≤ℵ 1 .The following problems were raised earlier.Question A6. Is it consistent that every ℵ 1 -collectionwise normal Moorespace is metrizable?Aspaceisℵ 1 -collectionwise normal if any discrete collection of size ℵ 1 canbe separated. In Tall [19∞b], assuming the consistency of a huge cardinal,I proved it consistent that ℵ 1 -collectionwise normal Moore spaces of size ≤ℵ 2 are metrizable. Assuming a not unreasonable axiom the consistency ofwhich is, however, not currently known to follow from the usual large cardinalaxioms, the cardinality restriction can be removed.Question C1. Is it consistent (assuming large cardinals) that every firstcountable ℵ 1 -collectionwise Hausdorff space is collectionwise Hausdorff?This question is due to Fleissner. Again, in the Lévy model, the propositionholds for spaces of local cardinality ≤ℵ 1 (Shelah [1977]). The questionhere is whether countably closed forcing can separate an unseparated discretecollection in a first countable space.Question D1. Is it consistent (assuming large cardinals) that every firstcountable weakly ℵ 1 -collectionwise Hausdorff space is weakly collectionwiseHausdorff?In Tall [19∞b], from the consistency of a huge cardinal I proved the consistencyof first countable weakly ℵ 1 -collectionwise Hausdorff spaces beingweakly ℵ 2 -collectionwise Hausdorff. Using an axiom the consistency of whichis not known to follow from the usual large cardinality axioms—but whichis considerably weaker than one previously alluded to—and a result of Watson[19∞], I can indeed get from ℵ 1 to all larger cardinals.Daniels [1988] obtained a first countable weakly ℵ 1 -collectionwise Hausdorffspace that is not weakly ℵ 2 -collectionwise Hausdorff, assuming MA plus2 ℵ0 = ℵ 2 .Question F2. Is there a (real) example of a first countable space X such 61. ?that X × (ω 1 +1)is normal, but X is not paracompact?The hypothesis that X × (ω 1 + 1) is normal is equivalent to X being normaland ℵ 1 -paracompact (Kunen, see Przymusiński [1984]). (A space isκ-paracompact if every open cover of size ≤ κ has a locally finite open refinement.)A non-reflecting stationary set of ω-cofinal ordinals in ω 2 is again