OPEN PROBLEMS IN TOPOLOGY

§C] Collectionwise Hausdorff Problems 25Question B2. Is there a consistent example of a locally compact normal 43. ?metacompact space that’s not paracompact?Under V = L (or indeed in any model in which normal spaces of character≤ℵ 1 are collectionwise Hausdorff) there is no such example (Watson [1982]).In ZFC there is none such that for each open cover U there is an n ∈ ω suchthat U has a point n-refinement (Daniels [1983]).Question B3. Is there a consistent example of a locally compact locally 44. ?connected normal space that’s not collectionwise normal?This problem is due to Nyikos. The only connection I know between localconnectivity and collectionwise normality is that locally compact locallyconnected perfectly normal spaces are collectionwise normal with respect tosubmetacompact closed sets (Alster and Zenor [1976], or see Tall [1984]).Question B4. Is it consistent that normal k-spaces are collectionwise nor- 45. ?mal?k-spaces are precisely the quotients of locally compact spaces. Partial resultshave been achieved by Daniels [19∞].Question B5. Is it consistent without large cardinals that normal manifolds 46. ?are collectionwise normal?Nyikos noted that a weakly compact cardinal suffices (Nyikos [1983]), orsee Tall [1982]. Rudin obtained a counterexample from ♦ + (Rudin [19∞]).This problem is related to A5 above, since the components have size ≤ 2 ℵ0 .C. Collectionwise Hausdorff ProblemsQuestion C1. Is it consistent (assuming large cardinals) that every first 47. ?countable ℵ 1 -collectionwise Hausdorff space is collectionwise Hausdorff?This is discussed in §F below.Question C2. Suppose κ is a singular strong limit and X is a normal space 48. ?of character less than κ. Suppose X is λ-collectionwise Hausdorff for all λ