OPEN PROBLEMS IN TOPOLOGY

10 Dow / Dow’s Questions [ch. 1? 17.Question 17. Is every compact sequential space of character (or cardinality)ω 1 hereditarily α-realcompact?This question is posed in Dow [1988a]. Nyikos defines a space to be α-realcompact if every countably complete ultrafilter of closed sets is fixed.ReferencesBalogh, Z.[1989] On compact Hausdorff spaces of countable tightness. Proc. Amer.Math. Soc., 105, 755–764.Baumgartner, J. E. and M. Weese.[1982] Partition algebras for almost-disjoint families. Trans. Amer. Math. Soc.,274, 619–630.Bell, M. and K. Kunen.[1981] On the pi-character of ultrafilters. C. R. Math. Rep. Acad. Sci. Canada,3, 351–356.Comfort, W. W., N. Hindman, and S. Negrepontis.[1969] F ′ -spaces and their products with P -spaces. Pac. J. Math., 28,459–502.van Douwen, E. K.[1981] Remote points. Diss. Math., 188, 1–45.van Douwen, E. K. and J. van Mill.[1980] Subspaces of basically disconnected spaces or quotients of countablycomplete Boolean Algebras. Trans. Amer. Math. Soc., 259, 121–127.Dow, A.[1982] Some separable spaces and remote points. Can. J. Math., 34,1378–1389.[1983a] CH and open subspaces of F -spaces. Proc.Amer.Math.Soc., 89,341–345.[1983b] Co-absolutes of βÆ \ Æ. Top. Appl., 18, 1–15.[1983c] On F -spaces and F ′ -spaces. Pac. J. Math., 108, 275–284.[1983d] Products without remote points. Top. Appl., 15, 239–246.[1983e] Remote points in large products. Top. Appl., 16, 11–17.[1984a] The growth of the subuniform ultrafilters on ω 1. Bull. Greek Math.Soc., 25, 31–51.[1984b] On ultrapowers of Boolean algebras. Top. Proc., 9, 269–291.[1985] Good and OK ultrafilters. Trans. Amer. Math. Soc., 290, 145–160.[1987] Some linked subsets of posets. Israel J. Math., 59, 353–376.[1988a] A compact sequential space. to appear in Erdős volume.[1988b] More remote points. unpublishable manuscript.[1988c] PFA and ω1. ∗ Top. Appl., 28, 127–140.[1989] A separable space with no remote points. Trans. Amer. Math. Soc.,312, 335–353.