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Search: All articles in the CJM digital archive with keyword forcing

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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 2016 (vol 69 pp. 502)

Fischer, Vera; Mejia, Diego Alejandro
Splitting, Bounding, and Almost Disjointness Can Be Quite Different
We prove the consistency of $$ \operatorname{add}(\mathcal{N})\lt \operatorname{cov}(\mathcal{N}) \lt \mathfrak{p}=\mathfrak{s} =\mathfrak{g}\lt \operatorname{add}(\mathcal{M}) = \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a} =\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c} $$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values.

Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-names
Categories:03E17, 03E35, 03E40

3. CJM 2013 (vol 66 pp. 303)

Elekes, Márton; Steprāns, Juris
Haar Null Sets and the Consistent Reflection of Non-meagreness
A subset $X$ of a Polish group $G$ is called Haar null if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb R$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszyński and Burke-Miller.

Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real
Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

4. CJM 2012 (vol 64 pp. 1182)

Tall, Franklin D.
PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$
Extending the work of Larson and Todorcevic, we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$ is a coherent Souslin tree.

Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinal
Categories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65

5. CJM 2009 (vol 61 pp. 604)

Hart, Joan E.; Kunen, Kenneth
First Countable Continua and Proper Forcing
Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight $\aleph_1$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Keywords:connected space, Continuum Hypothesis, proper forcing, irreducible map
Categories:54D05, 03E35

6. 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

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