SPM Bulletin 32

Dear colleague, Next year, our mutual friend Ljubiša Kočinac turns 65. For this occasion, I am organizing the IV Workshop on Coverings, Selections and Games in Topology. Ljubisa Kocinac initiated and started this series of conferences in 2002, Lecce, Italy.

The workshop will take place at the Department of Mathematics, Seconda Universit di Napoli, Caserta, Italy.

Tentative time 2.3. Further information. Each talk will last about 30 minutes.

Of course, in a period of financial cuts we do not know up to now the support that we can offer to participants, the amount of registration fee, etc. This is indeed a very preliminary report, written to circulate this important and happy news. We would appreciate your forwarding this information to anyone who may be interested in attending this conference.

On the behalf of the organizing committee, I hope to see you in Caserta. (1) If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BC ℵ 1 holds. (2) If it is consistent that BC ℵ 1 holds, then it is consistent that there is an inaccessible cardinal.

(3) If it is consistent that there is a 1-inaccessible cardinal with ω inaccessible cardinals above it, then ¬BC ℵω + (∀n < ω)BC ℵn is consistent. (4) If it is consistent that there is a 2-huge cardinal, then it is consistent that BC ℵω holds. (5) If it is consistent that there is a 3-huge cardinal, then it is consistent that BC κ holds for a proper class of cardinals κ of countable cofinality. http://arxiv.org/abs/1107.5383 Fred Galvin and Marion Scheepers 3.6. The topology of ultrafilters as subspaces of 2 ω . Using the property of being completely Baire, countable dense homogeneity and the perfect set property we will be able, under Martin’s Axiom for countable posets, to distinguish non-principal ultrafilters on ω up to homeomorphism. Here, we identify ultrafilters with subpaces of 2 ω in the obvious way. Using the same methods, still under Martin’s Axiom for countable posets, we will construct a non-principal ultrafilter U ⊆ 2 ω such that U ω is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. Finally, we will give some partial results about the relation of such topological properties with the combinatorial property of being a P-point. http://arxiv.org/abs/1108.2533 Andrea Medini and David Milovich 3.7. Another note on the class of paracompact spaces whose product with every paracompact space is paracompact. Abstract. The paper contains the following two results:

(1) Let X be a paracompact space and M be a metric space such that X can be embedded in M ℵ 1 in such a way that the projections of X onto initial countably many coordinates are closed. Then the product X × Y is paracompact for every paracompact space Y if and only if the first player of the G(DC, X) game, introduced by Telgarsky, has a winning strategy. ( 2) If X is paracompact space, Y is a closed image of X and the first player of the G(DC, X) game has a winning strategy, then also the first player of the G(DC, Y ) game has a winning strategy. K. Alster 3.8. On paracompactness in the Cartesian products and the Telgarsky’s game. Let X be a paracompact space and M be a metric space such that X can be embedded in M ℵ 1 in such a way that the projections p α : X → M α are closed at every x ∈ X, and p -1 α p α (x) is clopen for all x ∈ X. Then the product X × Y is paracompact for every paracompact space Y if and only if the first player of the G(DC, X) game, introduced by Telgarsky, has a winning strategy.

K. Alster 3.9. Elementary chains and compact spaces with a small diagonal. It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.

http://arxiv.org/abs/1106.5116 Daniel T. Soukup, Paul J. Szeptycki 3.5. Borel’s Conjecture in Topological Groups. We introduce a natural generalization of Borel’s Conjecture. For each infinite cardinal number κ, let BC κ denote this generalization. Then BC ℵ 0 is equivalent to the classical Borel conjecture. We obtain the following consistency results:

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