Proposition 9

Every Hausdorff quotient group of $\mathbf{R}^n$ is isomorphic to a product group $\mathbf{R}^h \times T^k$ ($0 \leq h + k \leq n$).

The product space $T^n$ (and, by abuse of language, the topological group $T^n$) is called the n-dimensional torus; by Proposition 4 of Chapter V, § 1, no. 2, $T^n$ is compact, connected and locally connected.

Furthermore, if $C$ denotes a closed cube of side 1 in $\mathbf{R}^n$, $T^n$ is homeomorphic to the quotient space of $C$ by the equivalence relation " $x_i \equiv y_i \pmod{1}$ ($1 \leq i \leq n$)" between the points $x = (x_i)$ and $y = (y_i)$ of $C$. Thus $T^n$ is formed from the cube $C$ by "identification of opposite faces".