Example 4.2.84D234E8193AC

Proof. A similarity $x\mapsto a+tx$ with $t>0$ preserves upper semicontinuity and sphere means, and hence maps polar sets to polar sets. It therefore suffices to find one uncountable compact polar subset of $\mathbb R^ d$. For $d=2$ we take the Cantor set $K$ of proposition 4.1.12. It is compact and polar, and it is uncountable because it carries an atom-free probability measure.

For $d>2$, write $x=(z,y)\in \mathbb R^2\times \mathbb R^{d-2}$. By lemma 4.2.7, applied with $k=2$ to the subharmonic function $p$ of proposition 4.1.12, the function $P(x)=p(z)$ is subharmonic on $\mathbb R^ d$. It is finite when $z\notin K$ and equals $-\infty $ on $K\times \mathbb R^{d-2}$. Hence $K\times \{ 0\} $ is an uncountable compact polar subset of $\mathbb R^ d$.

Now let $\Omega \neq \varnothing $, and let $K'\subset \Omega $ be an uncountable compact polar set. By proposition 4.2.6, $K'$ is finely closed, and its induced fine topology is discrete. A closed subspace $F$ of a Lindelöf space $X$ is Lindelöf: extend the members of an open cover of $F$ to open subsets of $X$, add $X\setminus F$, and pass to a countable subcover. On the other hand, the cover of an uncountable discrete space by its singletons has no countable subcover. Hence the fine topology on $\Omega $ is not Lindelöf. Nor is it second countable, because a second countable space is Lindelöf: given an open cover and a countable basis, choose, for each basis element contained in some member of the cover, one such member; these countably many members cover the space. $\square$