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one compact topology on such a group? We give in ?3 a necessary and sufficient condition that the topology be unique in the case of a connected group. Our considerations proceed from a general structure theorem given in ?1. In ?4 we give a structural theorem for compact, connected, abelian groups. The relations between compactness and connectedness arise constantly in the above arguments and this relation is definitely established in ?2. I would like to take this opportunity to express my appreciation to Dr. Ky Fan whose generous assistance made this work possible. 1. In this section we develop a structural theorem for compact, connected groups. We include a complete proof of the necessary lemmas though the arguments are standard. According to [6, p. 282] a compact, connected Lie group G can be ex

Uniqueness of the topology in certain compact groups