OPEN PROBLEMS IN TOPOLOGY

§D] Weak Separation Problems 27may or may not be disjoint from f(Y − Z). Since X is normal, there will besome f for which these sets are disjoint. In general, one would expect 2 |Y |basic functions would be needed to witness the normality of (the 2 |Y | subsetsof) Y .InTall [1981] I proved for Y ’s of cardinality ℵ 1 :D.1. Theorem.(a) If ≤ℵ 1 functions witness the normality of Y ,thenY is separated.(b) Assuming a generalized Martin’s Axiom (e.g. BACH), if < 2 ℵ1 functionswitness the normality of Y ,thenY is separated.(c) If 2 ℵ0 < 2 ℵ1 and < 2 ℵ1 functions witness the normality of Y ,thenthere is an uncountable separated subset of Y .Question D3. Is it consistent that there is a space X and a closed discrete 52. ?Y such that < 2 ℵ1 (better, < 2 ℵ0 ) functions witness the normality of Y , but(every uncountable Z ⊆) Y is not separated?Question D4. Is CH equivalent to the assertion that whenever < 2 ℵ0 53. ?functions witness the normality of Y , Y is separated?Question D5. Does 2 ℵ0 < 2 ℵ1 imply that assertion? 54. ?(See Watson [1985]) Steprāns and Watson proved that 2 ℵ0 < 2 ℵ1 ≤ℵ ω1implies countably paracompact separable spaces are collectionwise normal.(Note weakly collectionwise Hausdorff implies collectionwise normal for separablespaces.) Of course 2 ℵ0 < 2 ℵ1 suffices if we replace countable paracompactnessby normality.Question D6. Does 2 ℵ0 < 2 ℵ1 imply countably paracompact separable (first 55. ?countable?) spaces are collectionwise normal?Both this and Problem D5 are related to the following long-open hardproblem, which I believe is due to Laver.Question D7. Is it consistent that there is an F⊆ ω1 ω,with|F| < 2 ℵ1 , 56. ?such that F dominates all functions from ω 1 to ω?There is no such family if cf(2 ℵ0 ) < min(2 ℵ1 , ℵ ω1 ), while the existence ofsuch a family implies the existence of a measurable cardinal in an inner model(Steprāns [1982], Jech and Prikry [1984]).