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Search: MSC category 03E50 ( Continuum hypothesis and Martin's axiom [See also 03E57] )

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1. CJM 2012 (vol 64 pp. 1378)

Raghavan, Dilip; Steprāns, Juris
On Weakly Tight Families
Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira, who applied it to the Katétov order on almost disjoint families.

Keywords:maximal almost disjoint family, cardinal invariants
Categories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

2. CJM 2005 (vol 57 pp. 1139)

Burke, Maxim R.; Miller, Arnold W.
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
We prove that it is relatively consistent with $\ZFC$ that in any perfect Polish space, for every nonmeager set $A$ there exists a nowhere dense Cantor set $C$ such that $A\cap C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.

Keywords:Property of Baire, Lebesgue measure,, Cantor set, oracle forcing
Categories:03E35, 03E17, 03E50

3. CJM 1997 (vol 49 pp. 1089)

Burke, Maxim R.; Ciesielski, Krzysztof
Sets on which measurable functions are determined by their range
We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU)
Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50

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