Artur Hideyuki Tomita

Countably compact groups from a selective ultrafilter

We prove that the existence of a selective ultrafilter on ω implies the existence of a countably compact group without non-trivial convergent sequences all of whose powers are countably compact. Hence, by using a selective ultrafilter on ω, it is possible to construct two countably compact groups without non-trivial convergent sequences whose product is not countably cornpact.

A group under MAcountable whose square is countably compact but whose cube is not

Abstract We show under MAcountable the existence of a countable subgroup E of 2 c such that the group H generated by E and G = {x ϵ 2 c } : supp x is bounded in c is a group as in the title. We also show under MAcountable that for each k ϵ N there exists a countable family {{E n : n ϵ N }} of countable subgroups of 2 c such that if Hn = En + G, then for each subset F of N of size k, ΠnϵFHn is incountable compact, while for each subset F of N of size k + 1, ΠnϵFHn is not countably compact.

Extreme selections for hyperspaces of topological spaces

Abstract We study properties of Hausdorff spaces X which depend on the variety of continuous selections for their Vietoris hyperspaces F (X) of closed non-empty subsets. Involving extreme selections for F (X) , we characterize several classes of connected-like spaces. In the same way, we also characterize several classes of disconnected-like spaces, for instance all countable scattered metrizable spaces. Further, involving another type of selections for F (X) , we study local properties of X related to orderability. In particular, we characterize some classes of orderable spaces with only one non-isolated point.

Two countably compact topological groups: One of size ℵ_{} and the other of weight ℵ_{} without non-trivial convergent sequences

E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in ZFC. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is w 1 < c. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is N w .

SELECTIONS GENERATING NEW TOPOLOGIES

Every (continuous) selection for the non-empty 2-point subsets of a space

Baire spaces, Tychonoff powers and the Vietoris topology

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On the number of countably compact group topologies on a free Abelian group

naturally defines an interval-like topology on

Small cardinals and the pseudocompactness of hyperspaces of subspaces of βω

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FORCING COUNTABLY COMPACT GROUP TOPOLOGIES ON A LARGER FREE ABELIAN GROUP

. In the present paper, we demonstrate that, for a second-countable zero-dimensional space

On finite powers of countably compact groups

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