On powers of countably pracompact groups

Abstract

In 1990, Comfort asked: is there, for every cardinal number α≤2c, a topological group G such that Gγ is countably compact for all cardinals γ<α, but Gα is not countably compact? A similar question can also be asked for countably pracompact groups: for which cardinals α is there a topological group G such that Gγ is countably pracompact for all cardinals γ<α, but Gα is not countably pracompact? In this paper we construct such group in the case α=ω, assuming the existence of c incomparable selective ultrafilters, and in the case α=κ+, with ω≤κ≤2c, assuming the existence of 2c incomparable selective ultrafilters. In particular, under the second assumption, there exists a topological group G so that G2c is countably pracompact, but G(2c)+ is not countably pracompact, unlike the countably compact case.