Vol. 57, No. 2, 1975

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Some applications of ultrafilters in topology

John Norman Ginsburg and Victor Harold Saks

Vol. 57 (1975), No. 2, 403–418

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.

Mathematical Subject Classification 2000
Primary: 54D35
Received: 2 April 1974
Revised: 20 January 1975
Published: 1 April 1975
John Norman Ginsburg
Victor Harold Saks