Exercise 15, § 5

Let $X$ be the subset of the plane $\mathbf{R}^2$ consisting of the line $D = \{0\} \times \mathbf{R}$ and the points $(1/n, k/n^2)$, where $n$ runs through the set of integers $> 0$ and $k$ runs through the set of all rational integers. a) For each point $(0, y)$ of $D$ and each integer $n > 0$, let $T_n(y)$ be the set of points $(u, v) \in X$ such that $u \leqslant 1/n$ and $|v - y| \leqslant u$. Show that if we take as a fundamental system of neighbourhoods of each point $(0, y)$ the set of sets $T_n(y)$, and as a fundamental system of neighbourhoods of each other point of $X$ the set consisting of this point alone, then we have defined a topology $\mathscr{T}$ on $X$, which is finer than the topology induced by that of $\mathbf{R}^2$, and with respect to which $X$ is locally compact. b) Show that $X$ is the union of a countable family of metrizable closed subspaces and that every point of $X$ has a metrizable compact neighbourhood, but that $X$ is not normal. [Let $A$ be the set of all points $(0, y)$ where $y$ is rational, and let $B = D - A$; show that every neighbourhood $U$ of $A$ in $X$ meets every neighbourhood $V$ of $B$ in $X$; for this, consider for each $n$ the set $B_n$ of points $y \in B$ such that $T_n(y) \subset V$, and note that the set $B$ is not meagre in $\mathbf{R}$.]