Exercise 26, § 8

Let $(\theta_n)$ be a sequence of irrational numbers which is dense in the interval $I = [0, 1]$ of $\mathbf{R}$. Let $I_n$ be the quotient space of $I$ obtained by identifying 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 $I$; then the restriction of $\varphi_n$ to $X$ is a bijection of $X$ onto $\varphi_n(X)$. Let $\mathfrak{T}_n$ be the inverse image under this bijection of the topology induced on $\varphi_n(X)$ by the topology of $I_n$. Show that the greatest lower bound of the decreasing sequence $(\mathfrak{T}_n)$ of Hausdorff topologies on $X$ is not a Hausdorff topology.*