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1. 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 BartoszyÅski and Burke-Miller. Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic realCategories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

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 HruÅ¡Ã¡k and GarcÃ­a Ferreira, who applied it to the KatÃ©tov order on almost disjoint families. Keywords:maximal almost disjoint family, cardinal invariantsCategories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65

3. 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 cardinalCategories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65

4. CJM 2012 (vol 65 pp. 485)

Bice, Tristan Matthew
 Filters in C*-Algebras In this paper we analyze states on C*-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison-Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Lastly we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally we show that consistently all towers on the natural numbers remain towers under this embedding. Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towersCategories:46L03, 03E35

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 mapCategories: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 forcingCategories:03E35, 03E17, 03E50

7. CJM 2005 (vol 57 pp. 471)

Ciesielski, Krzysztof; Pawlikowski, Janusz
 Small Coverings with Smooth Functions under the Covering Property Axiom In the paper we formulate a Covering Property Axiom, \psmP, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Stepr\={a}ns. \begin{compactenum}[\rm(a)~~] \item There exists a family $\F$ of less than continuum many $\C^1$ functions from $\real$ to $\real$ such that $\real^2$ is covered by functions from $\F$, in the sense that for every $\la x,y\ra\in\real^2$ there exists an $f\in\F$ such that either $f(x)=y$ or $f(y)=x$. \item For every Borel function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^1$'' functions ({\em i.e.,} differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$. \item For every $n>0$ and a $D^n$ function $f\colon\real\to\real$ there exists a family $\F$ of less than continuum many $\C^n$ functions whose graphs cover the graph of $f$. \end{compactenum} We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevski\v{\i}. Keywords:continuous, smooth, coveringCategories:26A24, 03E35

8. 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