Proposition 4

The torus $\mathbf{T}$ is homoeomorphic to the quotient space of any closed interval of $\mathbf{R}$ of the form $[a, a+1]$ obtained by identifying the end-points of this interval; it is compact, connected and locally connected.

Every $x \in \mathbf{R}$ is congruent $(\bmod\,1)$ to some number in the interval $[a, a+1]$, namely to $x - [x-a]$; hence $\mathbf{T}$ is the image of this interval under the canonical mapping $\varphi$ of $\mathbf{R}$ onto $\mathbf{R}/\mathbf{Z}$, and is therefore compact and connected (Chapter I, § 9, no. 4, Theorem 2, and § 11, no. 2, Proposition 4). On the other hand, two distinct elements of the interval $[a, a+1]$ are congruent $(\bmod\,1)$ only if they are the end-points of the interval; from the compactness of $\mathbf{T}$ it therefore follows that $\mathbf{T}$ is homeomorphic to the quotient space of $[a, a+1]$ obtained by identifying the end-points (Chapter I, § 9, no. 4, Theorem 2, Corollary 4, and § 10, no. 4, Proposition 8). Finally, $\mathbf{Z}$ being a discrete subgroup of $\mathbf{R}$, $\mathbf{T} = \mathbf{R}/\mathbf{Z}$ is locally isomorphic to $\mathbf{R}$ (Chapter III, § 2, no. 6, Proposition 19) and in particular is locally connected (this last fact is also a consequence of Chapter I, § 11, no. 6, Proposition 12).