Exercise 4, § 2

Soit $D = [0,1]\cap (\mathbf{R}-\mathbf{Q})$.

a) Let $\varphi$ be a mapping from D into $[0,1]$ and let X be the complement in $[0,1]^2$ of the set formed by the $(x, \varphi (x))$ for $x\in D$. Prove that X is connected.

b) Prove that there exists a bijective mapping Φ from D onto the set of paths with origin $(0,0)$ and endpoint $(1,1)$ in $[0,1]^2$. For every $x\in D$, let $\psi (x)$ be a point of abscissa $x$ on the path $\Phi (x)$. Let then Y be the complement in $[0,1]^2$ of the set formed by the $\psi (x)$ for $x\in D$. Prove that the space Y is connected, locally connected but is not connected by arcs.