On ordinal-metric intersection topologies (Q1076358)

Following G. M. Reed the author considers the class C of all spaces X of size \(\omega_ 1\) for which there exist a separable metric topology \(T_ 1\) on the set X and an order topology \(T_ 2\) induced by some well-order of X of the type \(\omega_ 1\) such that \(\{\) \(U\cap V:\) U is open in \(T_ 1\) and V is open in \(T_ 2\}\) is a base of the topology of X. The author investigates normality and perfectness \((=\) all closed sets are \(G_{\delta}'s)\) in the class C and presents theorems which clear up connections between these properties. He proves that there are no spaces in C which are both perfect and normal. Under MA\(+\neg CH\) all spaces in C are perfect since separable metric spaces of size \(\omega_ 1\) are Q- sets. Hence, under \(MA+\neg CH\) all spaces in C are nonnormal. Under CH the situation is almost reversed: the author proves that for every \(X\subset C\) there exists a club set \(D\subset X\) such that D is normal. Since C is closed under club subsets, this gives a strengthening of a theorem of Reed. It is not known if a model of ZFC exists in which all members of C are neither perfect nor normal. But in this paper the author proves that if \(\omega_ 1\) Cohen reals are added to a model of \(MA+\neg CH\), then in the extension the class C will contain both perfect and normal spaces.