Displaying similar documents to “On π -caliber and an application of Prikry’s partial order”

Cardinal inequalities implying maximal resolvability

Marek Balcerzak, Tomasz Natkaniec, Małgorzata Terepeta (2005)

Commentationes Mathematicae Universitatis Carolinae

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We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space X is maximally resolvable provided that for a dense set X 0 ⊂ X and for each x ∈ X 0 the π -character of X at x is not greater than the dispersion character of X . On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.

Nowhere dense subsets and Booth's Lemma

Viacheslav I. Malykhin (1996)

Commentationes Mathematicae Universitatis Carolinae

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The following statement is proved to be independent from [ LB + ¬ CH ] : ( * ) Let X be a Tychonoff space with c ( X ) ≤ ℵ 0 and π w ( X ) < ℭ . Then a union of less than ℭ of nowhere dense subsets of X is a union of not greater than π w ( X ) of nowhere dense subsets.

Perfect sets and collapsing continuum

Miroslav Repický (2003)

Commentationes Mathematicae Universitatis Carolinae

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Under Martin’s axiom, collapsing of the continuum by Sacks forcing 𝕊 is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if 𝔡 = 𝔠 proving that under this hypothesis there are no small uncountable maximal antichains in 𝕊 . We also construct a partition of ω 2 into 𝔠 perfect sets which is a maximal antichain in 𝕊 and show that s 0 -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.