A Countably Compact Topological Group H Such that H × H is not Countably Compact

Abstract

Using MAcountable we construct a topological group with the properties mentioned in the title. 0. Introduction In this paper we construct, assuming Martin's Axiom for countable posets (MAcountable), a countably compact topological group 77 for which 77 is not countably compact. The existence of such a group under MA was announced by van Douwen in [vD] but was never published. Our construction is like van Douwen's construction in [vD] in that our group is a subgroup of c2 and that we construct the points of our group in an induction of length c. But whereas van Douwen used MA to construct a countably compact subgroup of c2 without convergent sequences and then constructed two countably compact subgroups of this group with a countable intersection, we construct our group all at once. Moreover, our group 77 can be written as G + D with D a countable subgroup and G an eobounded subgroup, so that our group has many convergent sequences. Also van Douwen needed MA for certain uncountable posets, we get by using MAcountable only. The interest in examples such as the ones in the present paper comes from the fact that by Comfort and Ross [CR] the product of an arbitrary family of pseudocompact topological groups is pseudocompact. (For more information see [vD].) This paper is organized as follows: §1 contains some definitions and preliminaries. In §2 we use our technique to construct two countably compact subgroups 770 and 77j of c2 such that 770 x Hx is not countably compact, and in §3 we construct our main example 77. The reason for doing this is that the construction in §3 is rather involved notationwise so that it may be useful to have a more transparant version available. Also, in §3, we essentially construct a countably compact subgroup 77 of £2x'2 which contains (modified

Countably compact groups from a selective ultrafilter

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A solution to Comfort's question on the countable compactness of powers of a topological group

“…MA countable is equivalent to the statement "the intersection of fewer than c dense open sets of the real line is dense" (see [15]). MA implies MA countable and MA countable implies the existence of selective ultrafilters.…”

“…Hart and van Mill [15] showed in 1991 that there exists a countably compact topological group whose square is not countably compact under MA countable . The example contains non-trivial convergent sequences as it makes use of ω-bounded groups in the construction.…”

“…(1) Hart and van Mill [15]: There exists a countably compact group whose square is not countably compact from MA countable . (2) Ginsburg and Saks [13]: If the 2 c th power of a topological space X is countably compact then every power of X is countably compact.…”

Two countably compact topological groups: One of size $\aleph _\omega $ and the other of weight $\aleph _\omega $ without non-trivial convergent sequences

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