OPEN PROBLEMS IN TOPOLOGY

8 Dow / Dow’s Questions [ch. 1? 6.Question 6. Does every Parovichenko space have a c×c-independent matrix?This is a technical question which probably has no applications but I findit interesting. A Parovichenko space is a compact F-space of weight c inwhich every non-empty G δ has infinite interior. The construction of a c × c-independent matrix on P(ω) usesheavilythefactthatω is strongly inaccessible,see Kunen [1978]. In Dow [1985] it is shown that each Parovichenkospace has a c × ω 1 -independent matrix and this topic is also discussed inDow [1984b, 1984a].? 7.Question 7. Is cf(c) =ω 1 equivalent to the statement that all Parovichenkospaces are co-absolute?It is shown in Dow [1983b] that the left to right implication holds.? 8.Question 8. Is there a clopen subset of the subuniform ultrafilters of ω 1whose closure in βω 1 is its one-point compactification?This is a desperate attempt to mention the notion and study of coherent sequences(Dow [1988c] and Todorčević [1989]). These may be instrumentalin proving that ω ∗ is not homeomorphic to ω ∗ 1 .? 9.Question 9. What are the subspaces of the extremally disconnected spaces?More specifically, does every compact basically disconnected space embed intoan extremally disconnected space?E. K. van Douwen and J. van Mill [1980] have shown that it is consistentthat not every compact zero-dimensional F-space embeds and it is shownin Dow and van Mill [1982] that all P-spaces and their Stone-Čech compactificationsdo. It is independent of ZFC whether or not open subspaces ofβN\N are necessarily F-spaces (Dow [1983a]). There are other F-spaces withopen subspaces which are not F-spaces. The references Dow [1982, 1983c]are relevant.? 10.Question 10. Find a characterization for when the product of a P-space andan F-space is again an F-space.A new necessary condition was found in Dow [1983c] and this had severaleasy applications. See also Comfort, Hindman and Negrepontis [1969]for most of what is known.? 11.Question 11.disconnected?Is the space of minimal prime ideals of C(βN \ N) basically