The main result of this paper is a structure theorem for compact connected groups, showing that every connected compact group \(G\) is the inverse limit of an inverse system \(\{(G_\alpha,p_{\alpha}^{\alpha+1}):\alpha<\vartheta\}\) of compact groups such that \(G_0\) is trivial and either \(p_\alpha^{\alpha+1}:G_{\alpha+1}\to G_\alpha\) has finite kernel or \(G_{\alpha+1}=G_\alpha\times H\), where \(H\) is a compact Lie group and \(p_\alpha^{\alpha+1}:G_\alpha\times H\to G_\alpha\) is the canonical projection. It is noted that the counterpart of this theorem holds for Abelian compact groups and for \(0\)-dimensional compact groups, while the group \(\mathbb T\rtimes \mathbb Z_2\) is given to show that this is not the case for compact groups that are not connected.
As an application of their main theorem, the authors prove that every compact group is supercompact. The equivalence of compactness and supercompactness for topological groups was announced by Mills in a seminar report; his proof is described in this paper.
Reviewer’s remark: The definition of simple compact Lie group used in this paper differs from the usual one; indeed, in this paper the torus \(\mathbb T\) is intended to be a simple compact Lie group.