Let \(\mathscr{O}\) be the collection of open sets (topology) in \(\mathbb{R}^{n}\) and let \(A \subset \mathbb{R}^{n}\) be an arbitrary subset. We can introduce a topology \(\mathscr{O}_{A}\) on \(A\) as follows: a set \(V \subset A\) is called open (relative to \(A\) ) if \(V=U \cap A\) for some \(U \in \mathscr{O}\). We write \(\mathscr{O}_{A}\) for the open sets relative to \(A\).

(i). Show that \(\mathscr{O}_{A}\) is a topology on \(A\), i.e. a family satisfying \(\left(\mathscr{O}_{1}ight)-\left(\mathscr{O}_{3}ight)\).

(ii). If \(A \in \mathscr{B}\left(\mathbb{R}^{n}ight)\), show that the trace \(\sigma\)-algebra \(A \cap \mathscr{B}\left(\mathbb{R}^{n}ight)\) coincides with \(\sigma\left(\mathscr{O}_{A}ight)\) (the