Vol. 19, No. 2, 1966

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Pseudocompact groups

Howard Joseph Wilcox

Vol. 19 (1966), No. 2, 365–379

This paper is divided into three sections. The first of these concerns itself with extending a result known for compact Abelian topological groups to arbitrary compact topological groups. The particular result is Theorem 1 of “Extensions of Haar Measure for Compact Connected Abelian Groups” by Gerald L. Itzkowitz (Bull. Am. Math. Soc., Vol 71, p. 152–156, 1965). The method used to extend this result is the following: the result is shown to hold for two special cases (when G is a compact o-dimensional group; and when G is a product of connected, compact, metric groups), and then it is proven that whenever the result holds for a closed normal subgroup J of a compact group G and also for the factor groups G∕J then the result holds for G itself. Using this fact together with the special cases and two known structure theorems yields the desired extension to arbitrary compact groups.

Section II uses the key result of §I (that many compact groups contain dense pseudocompact subgroups of small cardinality) to show that many Abelian compact groups have infinite compact subgroups which meet the dense pseudocompact subgroups mentioned above in only the identity element. A related result shows that many compact Abelian groups contain dense pseudocompact subgroups which are not countably compact. Counterexamples demonstrate that these results do not hold in general for arbitrary compact groups.

In §III we define the subset of metric elements of a locally compact group to be all those elements which have metric groups as the smallest closed subgroups containing them. We show that for infinite compact Abelian groups the collection M of metric elements is a dense pseudocompact subgroup. We also produce necessary and sufficient conditions for every element of a compact Abelian group to be a metric element. Finally, we show by counterexample that the collection M is not even a subgroup in general when G is nonAbelian.

Mathematical Subject Classification
Primary: 22.20
Received: 21 September 1965
Revised: 23 January 1966
Published: 1 November 1966
Howard Joseph Wilcox