Exercise 7, § 2

(a) Let X be the set [0; 1[. Let $\mathscr{T}_X$ be the set of subsets U of X satisfying the two properties:

(i) The intersection $U\cap ]0; 1[$ is open ;

(ii) If $0\in U$, there exists an integer $p\geqslant 1$ such that $1/n\in U$ for every integer $n\geqslant p$.

Exercise prove that $\mathscr{T}_X$ is a topology on X.

(b) Let Z be the set $[0,1[^2\times (\mathbf{Z}/2\mathbf{Z})$. Let $\mathscr{T}_Z$ be the set of subsets U of Z satisfying:

(i) The intersection of U with the complement of $\{(0,0)\} \times (\mathbf{Z}/2\mathbf{Z})$ is open in the space Z endowed with the product topology of $X\times X\times (\mathbf{Z}/2\mathbf{Z})$;

(ii) For every $a\in \mathbf{Z}/2\mathbf{Z}$ such that $(0,0, a)\in U$, there exists an integer $p\geqslant 1$ such that, for every couple $(m, n\surd )$ of natural integers satisfying$\surd$ inf($m, n$)$\geqslant p$, one has $(1/n,\surd 1/m, a)\in \surd U$ if $(n-m2)(m-n2)>0$, $(1/n,1/m, a+ 1)\in U$ if $(n-m2)(m-n2)<0$, $(0,1/m, a)\in U$ and $(1/n,0, a)\in U$.

Exercise prove that $\mathscr{T}_Z$ is a topology on Z.

(c) The sets X and Z are endowed with these topologies, and the space $X\times X$ with the product topology. Let $p: Z\rightarrow X\times X$ be the canonical projection. Exercise prove that $p$ makes Z a covering of degree 2 of $X\times X$.

(d) Exercise prove that $p$ admits no continuous section.

(e) Let $s: X\times X\rightarrow Z$ be the mapping given by $(x, y)\mapsto (x, y,0)$. Exercise prove that for every $x\in X$, the restriction of $s$ to $X\times \{x\}$ is continuous, analogously the restriction of $s$ to $\{x\} \times X$ is continuous.