Exercise 26, § 8

Let $(\theta_n)$ be a sequence of irrational numbers everywhere dense in the interval I = (0, 1) of $\mathbf{R}$. Let $I_n$ be the quotient space of I obtained by identifying with one another all the points 0, $\theta_1, \ldots, \theta_n$, and let $\varphi_n$ be the canonical mapping $I \to I_n$. Let X be the set of rational numbers contained in I; the restriction of $\varphi_n$ to X is a bijection of X onto $\varphi_n(X)$; let $\mathfrak{T}_n$ be the inverse image by this bijection of the topology induced on $\varphi_n(X)$ by the topology of $I_n$. Now prove that the greatest lower bound of the decreasing sequence $(\mathfrak{T}_n)$ of separated topologies on X is not separated.*