## Faculty Publications

# Resolvability In C.C.C. Generic Extensions

## Document Type

Article

## Keywords

Measurable cardinal, Monotonically ω1-resolvable, Resolvable

## Journal/Book/Conference Title

Commentationes Mathematicae Universitatis Carolinae

## Volume

58

## Issue

4

## First Page

519

## Last Page

529

## Abstract

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-resolvable in 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

Department of Mathematics

## Original Publication Date

1-1-2017

## DOI of published version

10.14712/1213-7243.2015.226

## Repository

UNI ScholarWorks, Rod Library, University of Northern Iowa

## Language

en

## Recommended Citation

Soukup, Lajos and Stanley, Adrienne, "Resolvability In C.C.C. Generic Extensions" (2017). *Faculty Publications*. 955.

https://scholarworks.uni.edu/facpub/955