Exercise 1, § 5

Let $\alpha \in \mathbf{R}-\mathbf{Q}$ be an irrational real number and let D be the line of $\mathbf{R}^2$ generated by the vector $(1, \alpha )$. Let $p$ be the canonical mapping of $\mathbf{R}^2$ onto $\mathbf{R}^2/\mathbf{Z}^2$. We set $B =p(D)$. a) Now prove that $(\overset{-1}{p}(B), p_B)$ is a covering of B.

b) Now prove that D is a connected component of $\overset{-1}{p}(B)$ but that $(D, p_B)$ is not a covering.

This shows that in Proposition 6 of I, p. 103, the assumption that B is locally connected cannot be removed. The mapping $p: D\rightarrow B$ also provides an example of a continuous bijective mapping such that $(D, p)$ is not a covering.