Journal of Lie Theory

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/G₀ of G modulo the identity component G₀ 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:
Theorem. 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.