Remark

Note that the canonical mapping $\varphi$ of $\mathbf{R}$ onto $T = \mathbf{R}/\mathbf{Z}$, restricted to the half-open interval $[a, a+1[$, is a continuous bijective mapping of this interval onto $T$; the inverse mapping is continuous at every point of $T$ other than $\varphi(a)$, but discontinuous at $\varphi(a)$. We shall sometimes identify the space $T$ with the interval $[a, a+1[$, endowed with the topology which is the inverse image under $\varphi$ of the topology of $T$ (Chapter I, § 1, no. 3); this topology is of course not the same as that induced on $[a, a+1[$ by the topology of $\mathbf{R}$.