OPEN PROBLEMS IN TOPOLOGY

§H] Real Line Problems 31Under CH, there is a first countable L-space (van Douwen, Tall andWeiss [1977]) and hence an example. If 2 ℵ0 < 2 ℵ1 , such an example wouldyield a first countable L-space (Tall [1974b]). A model in which there were nosuch space would both have to have no first countable L-space and yet have everyfirst countable normal space be collectionwise Hausdorff (Tall [1974b])—a very curious combination indeed!Question G5. Is it consistent with GCH that precaliber ℵ 1 implies precal- 67. ?iber ℵ ω+1 ?Spaces with precaliber ℵ 1 do have precaliber ℵ ω+1 if one assumes the axiomalluded to in §F (Tall [19∞b]).H. Real Line ProblemsThe rational sequence topology (see Steen and Seebach [1978]), the Pixley-Roy topology (see e.g., van Douwen [1977]), and the density topology (seee.g., Tall [1976a]) are all strengthenings of the usual topology on the realline. For the first two, there is a characterization of normal subspaces interms of their properties as sets of reals. By the same proof (Bing [1951],Tall [1977b]) as for the tangent disk space, a set X of reals is normal in therational sequence topology iff it’s a Q-set in the usual topology, while X isnormal in the Pixley-Roy topology iff it’s a strong Q-set (Rudin [1983]).Question H1. Characterize the normal subspaces of the density topology. 68. ?In Tall [1978] I obtained the following partial result:H.1. Theorem. If Y is a normal subspace of the density topology, thenY = S∪T ,whereS is generalized Sierpiński, T is a nullset such that Z ∩T = ∅for every nullset Z ⊆ S, every subset of Z is the intersection of Z with aEuclidean F σδ .(A set S of reals is generalized Sierpiński if its intersection with everynullset has cardinality less than continuum.) The closure referred to is inthe density topology, so that even if the converse were proved, the resultingcharacterization would not be quite satisfactory. On the other hand, underMA plus not CH, one can construct a generalized Sierpiński S (namely oneof outer measure 1) and a nullset T disjoint from S such that S ∪ T is notnormal (since |T | =2 ℵ0 ) and yet every null Z ⊆ S is a Q-set.