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.
