On the Component Factor Group G/G₀ of a Pro-Lie Group G

A pro-Lie group $G$ is a topological group such that $G$ is isomorphic to the projective limit of all quotient groups $G/N$ (modulo closed normal subgroups $N$) such that $G/N$ is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group $G_t:= G/G_0$ of $G$ modulo the identity component $G_0$ is compact. In this case it is straightforward that each Lie group quotient $G/N$ of $G$ has finitely many components. However, in spite of a comprehensive literature on pro-Lie groups, the following theorem, proved here, was not available until now: A pro-Lie group $G$ is almost connected if each of its Lie group quotients $G/N$ has finitely many connected components. The difficulty of the proof is the verification of the completeness of $G_t$.