Resolvability in c.c.c. generic extensions
Measurable cardinal, Monotonically ω1-resolvable, Resolvable
Commentationes Mathematicae Universitatis Carolinae
Every crowded space X is ω-resolvable in the c.c.c. generic extension V Fn(|X|,2) of the ground model.We investigate what we can say about λ-resolvability in c.c.c. generic extensions for λ > ω.A topological space is monotonically ω1-resolvable if there is a function f : X → ω1 such that(x ∈ X: f(x) ≥ α) ⊃dense X for each α < ω1.We show that given a T1 space X the following statements are equivalent:(1) X is ω1-resolvable in some c.c.c. generic extension;(2) X is monotonically ω1-resolvable;(3) X is ω1-resolvable in the Cohen-generic extension V Fn(ω1,2).We investigate which spaces are monotonically ω1-resolvable. We show that if a topological space X is c.c.c., and ω1 ≤ Δ(X) = |X| < ωω, where Δ(X) = min(|G|: G 6≠ ∅ open), then X is monotonically ω1-resolvable.On the other hand, it is also consistent, modulo the existence of a measurable cardinal, that there is a space Y with |Y | = Δ(Y ) = ℵω which is not monotonically ω1-resolvable.The characterization of ω1-resolvability in c.c.c. generic extension raises the following question: is it true that crowded spaces from the ground model are ω-resolvable in V Fn(ω,2)? We show that (i) if V = L then every crowded c.c.c. space X is ω-resolvablein V Fn(ω,2), (ii) if there are no weakly inaccessible cardinals, then every crowded space X is ω-resolvable in V Fn(ω1,2).Moreover, it is also consistent, modulo a measurable cardinal, that there is a crowded space X with |X| = Δ(X) = ω1 such that X remains irresolvable after adding a single Cohen real.
Department of Mathematics
Original Publication Date
DOI of published version
UNI ScholarWorks, Rod Library, University of Northern Iowa
Soukup, Lajos and Stanley, Adrienne, "Resolvability in c.c.c. generic extensions" (2017). Faculty Publications. 955.