Exercise 3, § 4

Let $B$ be the set of points $(x, y)$ of the square $[0,1]^2$ satisfying $x= 0$, or $x= 1$, or $y= 0$, or $y= 1$, or there exists $n\in \mathbf{N}$ such that $nx= 1$. Let $B_1$ and $B_2$ be the subsets of B formed by the points $(x, y)$ of B for which $y\leqslant \frac{1}{2}$ and $y\geqslant \frac{1}{2}$ respectively.

Construct a B-space $(E, p)$ which is not a covering but such that the $B_1$-space $(E_{B_1}, p_{B_1})$ and the $B_2$-space $(E_{B_2}, p_{B_2})$ are coverings.