On thin-tall scattered spaces

Superatomic Boolean algebras constructed from morasses

“…In [4] it was even proved that there exists an (co, a)-sBa for each a < C02, which in ZFC is the best we can hope for. In [4] it was even proved that there exists an (co, a)-sBa for each a < C02, which in ZFC is the best we can hope for.…”

“…We consider 4 Note that if < ,u < 4 < i, f E &u, g E @4, p,q E P with Yp,Yq C 0s, then pgof = (pf )g, and p < q implies pf < qf . We consider 4 Note that if < ,u < 4 < i, f E &u, g E @4, p,q E P with Yp,Yq C 0s, then pgof = (pf )g, and p < q implies pf < qf .…”

A Consistency Result on Thin-Tall Superatomic Boolean Algebras

“…(1) Juha'sz and Weiss proved in [2] that, for every ordinal a < 02, there exists an sBa B, such that ht(B,) = a and wd(Ba) = co. Then, by using the well-known fact that there is an almost disjoint family of 2W subsets of co, we obtain that 1 CH implies the existence of an sBa with exactly (o atoms and height a02. On the other hand, since the partial ordering P", is countably closed, we infer that forcing with P,, preserves CH (see [3, Theorem VII.6.…”

Cardinal sequences and Cohen real extensions

“…Thus, in particular, under CH there is no scattered T 3 space of height ω 2 and having only countably many isolated points. After I. Juhász and W. Weiss ( [5,Theorem 4]) had proved in ZFC that for every α < ω 2 there is an LCS space X with ht(X) = α and wd(X) = ω, it was a natural question if the existence of an LCS space of height ω 2 and width ω follows from ¬CH. This question was answered in the negative by W. Just who proved ([6, Theorem 2.13]) that if one blows up the continuum by adding Cohen reals to a model of CH then in the resulting generic extension there is no LCS space of height ω 2 and width ω.…”