Double session at Fields Institute (Menachem Magidor and Vera Fischer)
Place: Fields Institute, Room 210
Date: 31-October- 2014
Speaker 1 (from 12:30 to 13:30): Vera Fischer
Title: Definable Maximal Cofinitary Groups and Large Continuum
Abstract: A cofinitary group is a subgroup of the group of all permutations of the natural numbers, all non-identity elements of which have only finitely many fixed points. A cofinitary group is maximal if it is not properly contained in any other cofinitary group. We will discuss the existence of nicely definable maximal cofinitary groups in the presence of large continuum and in particular, we will see the generic construction of a maximal cofinitary group with a $\Pi^1_2$ definable set of generators in the presence of $2^\omega=\aleph_2$.
Speaker 2 (from 13:30 to 15:00): Menachem Magidor
Title: On compactness for being $\lambda$ collectionwise hausdorff
Abstract: A compactness property is the statement for a structure in a given class, if every smaller cardinality substructure has a certain property then the whole structure has this property. In this talk we shall deal with the compactness for the property of a topological space being collection wise Hausdorff. The space is X is said to be $\lambda$–collection wise Hausdorff ($\lambda$–cwH) if every closed discrete subset of X of cardinality less than $\lambda$ can be separated by a family of open sets. X is cwH if it is $\lambda$–cwH for every cardinal $\lambda$.
We shall deal with the problem of when $\lambda$–cwH implies cwH, or just when does $\lambda$–cwH implies $\lambda^+$–cwH. A classical example of Bing provides for every cardinal $\lambda$ a space $X_\lambda$ which is $\lambda$-cwH but not $\lambda^+$–cwH. So if we hope to get any level of compactness for the the property of being cwH, we have to restrict the class of spaces we consider. A fruitful case is the case where we restrict the local cardinality of the space. A motivating result is the construction by Shelah (using supercompact cardinal) of a model of Set Theory in which a space which is locally countable and which is $\omega_2$–cwH is cwH.
Can the Shelah result be generalized to larger cardinals , e.g. can you get a model in which for spaces which are locally of cardinality $\leq \omega_1$ and which are $\omega_3$-cwH are cwH? In general for which pair of cardinals $(\lambda, \mu)$ we can have models in which a space which is locally of cardinality $< \mu$ and which is $\lambda$–cwH are $\lambda^+$–cwH? In this lecture we shall give few examples where we get some ZFC theorems showing that for some pairs $(\lambda, \mu)$ compactness necessarily fails, and cases of pairs for which one can consistently have compactness for the property of being cwH.
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