In this paper, those
Hausdorff spaces, all of whose powers are countably compact, are characterized,
and partial results on the corresponding question for pseudocompactness
are obtained. The basic tool in this work is A. R. Bernstein’s concept of
𝒟-compactness. Sufficient conditions are found for every power of a topological
group to be countably compact. The maximal 𝒟-compact extension of a
completely regular space is constructed, and this procedure is used to construct
the unique 𝒟-compact group extension of a totally bounded topological
group. Additional product theorems for pseudocompact spaces are proved,
imposing conditions closely related to 𝒟-compactness on the factors, which
imply the pseudocompactness of the product. In the final section of the
paper, several theorems are proved which provide new examples of nontrivial
pseudocompact spaces. In particular, a homogeneous space is exhibited, all of whose
powers are pseudocompact, in which no discrete countable set has a cluster
point.
