English

On powers of countably pracompact groups

General Topology 2022-11-29 v1 Logic

Abstract

In 1990, Comfort asked: is there, for every cardinal number α2c\alpha \leq 2^{\mathfrak{c}}, a topological group GG such that GγG^\gamma is countably compact for all cardinals γ<α\gamma<\alpha, but GαG^\alpha is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals α\alpha is there a topological group GG such that GγG^{\gamma} is countably pracompact for all cardinals γ<α\gamma < \alpha, but GαG^{\alpha} is not countably pracompact? In this paper we construct such group in the case α=ω\alpha = \omega, assuming the existence of c\mathfrak{c} incomparable selective ultrafilters, and in the case α=κ+\alpha = \kappa^{+}, with ωκ2c\omega \leq \kappa \leq 2^{\mathfrak{c}}, assuming the existence of 2c2^{\mathfrak{c}} incomparable selective ultrafilters. In particular, under the second assumption, there exists a topological group GG so that G2cG^{2^\mathfrak{c}} is countably pracompact, but G(2c)+G^{(2^{\mathfrak{c}})^{+}} is not countably pracompact, unlike the countably compact case.

Keywords

Cite

@article{arxiv.2211.14598,
  title  = {On powers of countably pracompact groups},
  author = {Artur Hideyuki Tomita and Juliane Trianon-Fraga},
  journal= {arXiv preprint arXiv:2211.14598},
  year   = {2022}
}
R2 v1 2026-06-28T07:13:37.938Z