Proposition 7

In order that two connected unrollable topological groups be locally isomorphic, it is necessary and sufficient that their universal coverings be isomorphic topological groups.

A connected unrollable topological group is locally isomorphic to its universal covering (IV, p. 369, prop. 1); the condition is therefore sufficient. It is necessary by corollary 2 of proposition 2 (IV, p. 372), since the universal covering of a connected unrollable topological group is simply connected (IV, p. 379, prop. 6).