Proposition 10

The topological group $T^n$ is locally isomorphic to $R^n$.

For $T^n = (R/Z)^n$ is isomorphic to $R^n/Z^n$ (Chapter III, § 2, no. 9, Corollary to Proposition 26) and $Z^n$ is a discrete subgroup of $R^n$ (Chapter III, § 2, no. 6, Proposition 19).

It follows that the groups $R^p \times T^{n-p}$ are all locally isomorphic to $R^n$ ($0 \geq p \geq n$); in § 2, no. 2, we shall see that they are the only connected groups which have this property.