Exercise 24, § 8

In the interval $]0,1[$ of the rational line, let $A$ be the set of rational numbers of the form $k/2^n$, and $B$ the set of rational numbers of the form $k/3^n$ ($k$ an integer). Let $X$ be the space obtained by giving $]0,1[$ the topology generated by the open intervals $]\alpha,\beta[$ contained in $X$, and the sets $A$ and $B$. Show that $X$ is Hausdorff and semi-regular (Exercise 20) but not regular.