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

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1. CJM Online first

Dow, Alan; Tall, Franklin D.
Normality versus paracompactness in locally compact spaces
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.

Keywords:normal, paracompact, locally compact, countably tight, collectionwise Hausdorff, forcing with a coherent Souslin tree, Martin's Maximum, PFA(S)[S], Axiom R, moving off property
Categories:54A35, 54D20, 54D45, 03E35, 03E50, 03E55, 03E57

2. 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

3. 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

4. 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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