1. CJM Online first
|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 BartoszyÅski 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
2. CJM 2012 (vol 64 pp. 1182)
|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
3. CJM 2009 (vol 61 pp. 604)
|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
4. CJM 2005 (vol 57 pp. 1139)
|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