Exercise 4, § 4

Let $B$ be the subspace $\{1/n|n\in \mathbf{N}^*\} \cup \{0\}$ of $[0,1]$ and let $E = B\times \mathbf{N}$ $\{(0,0)\}$.

a) Exercise prove that E, endowed with the mapping pr$_1: E\rightarrow B$, is a trivial covering of B. b) Exercise prove that the canonical injection $E,\rightarrow B\times \mathbf{N}$ is a morphism of coverings of B but does not make E a covering of $B\times \mathbf{N}$.

c) Construct two B-spaces $(E_1, p_1)$ and $(E_2, p_2)$ and a B-morphism $f: E_1\rightarrow E_2$ such that $(E_1, f)$ and $(E_1, p_1)$ are coverings, but such that the mapping $p_2$ is not separated.

d) Exercise prove that the set U of pairs of the form $(1/n, n)$, for $n$ running through $\mathbf{N}^*$, is open and closed in $B\times \mathbf{N}$, but that $(U$, pr$_1|U)$ is not a covering of B.