Fine topology

Remarks

The fine topology is finer than the Euclidean topology, since every Euclidean open set is finely open by definition. A set $V$ is finely open if and only if, for every $x \in V$, the complement $\R^n \setminus V$ is thin at $x$. Example: let $V = \R^3 \setminus \set{k^{-1} e_1 : k \ge 1}$. The removed set is countable, hence a polar set, and a polar set is thin at every point. So $V$ is finely open, but it is not Euclidean open, because every ball about $0 \in V$ meets the removed points. Non-example: a closed half-space $H$ is not a fine neighbourhood of a point $x \in \partial H$, since the complementary open half-space is not thin at $x$.