Exercise 5, § 1

Let X be the subset of the plane $\mathbf{R}^2$ formed by the union of the line $D = \{0\} \times \mathbf{R}$ and the set of points $(1/n, k/n^2)$, where $n$ runs over the set of integers $> 0$, and $k$ over the set $\mathbf{Z}$ of rational integers.

a) For every point $(0, y)$ of D and every integer $n > 0$, let $T_n(y)$ be the set of points $(u, v)$ in X such that $u \leq 1/n$ and $|v - y| \leq u$. Show that if one takes as fundamental system of neighborhoods of each point $(0, y)$ in D the set of $T_n(y)$ ($n > 0$), and as fundamental system of neighborhoods of each of the other points of X the unique set reduced to that point, one defines on X a topology $\mathscr{T}$ for which X is a locally compact space that is not paracompact.

b) Let $\alpha$ be the numerical function on X that is equal to 0 on D and to $1/n^3$ at each of the points $(1/n, k/n^2)$. Show that, for every compact subset K of X, $\sum_{x \in K} \alpha(x)$ is finite; let $\mu$ be the positive measure on X defined by the masses $\alpha(x)$. Show that $\mu^*(D) = +\infty$ even though $\alpha(x) = 0$ on D. (If an open set U for $\mathscr{T}$ contains D, show that there exist an interval $a \leq y \leq b$ on D not reduced to a point, a set B dense in this interval (for the usual topology of $\mathbf{R}$), and an integer $n > 0$ such that, for every $y \in B$, one has $T_n(y) \subset U$; for this, one may make use of Baire’s theorem.)