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Resolvability in c.c.c. generic extensions

Lajos SoukupAdrienne Stanley — 2017

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 ω 1 -resolvable if there is a function f : X ω 1 such that { x X : f ( x ) α } d e n s e X for each α < ω 1 . We show that given a T 1 space X the following statements are equivalent: (1) X is ω 1 -resolvable in some c.c.c. generic extension; (2) X is monotonically ω 1 -resolvable; (3) X is ω 1 -resolvable in the Cohen-generic extension V Fn ( ω 1 , 2 ) ....

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