Corollary 1

If the $G_\alpha$ are compact and the $X_\alpha$ Hausdorff, then the conclusions of a) and b) are valid.

For the hypotheses of a) and b) are satisfied, since every closed subgroup of $G_\alpha$ is compact and $u_\alpha : s_\alpha \to s_\alpha \cdot x_\alpha$ is a continuous mapping of a compact space into a Hausdorff space.