Exercise 18, § 11

Let X be the real plane $\mathbf{R}^2$, and let R be the equivalence relation on X whose equivalence classes are the lines $y = \beta$ for $\beta > 0$ and the half-lines $x = \alpha, y \leq 0$ for all $\alpha \in \mathbf{R}$. Show that if X/R is regarded as a subset of $\mathfrak{B}_0(X)$, then the topology induced on X/R by $\mathfrak{T}_\Omega$ (§ 2, Exercise 7) is discrete, and that the topology induced on X/R by $\mathfrak{T}_\Phi$ is not discrete but makes X/R into a non-connected space. Show finally that X/R is connected in the quotient topology.*