91Ó°ÊÓ

The distance \(d\left(E_{1}, E_{2}\right)\) between the sets \(E_{1}, E_{2} \subset \mathbb{R}^{m}\) is the quantity $$ d\left(E_{1}, E_{2}\right):=\inf _{x_{1} \in E_{1}, x_{2} \in E_{2}} d\left(x_{1}, x_{2}\right) $$ Give an example of closed sets \(E_{1}\) and \(E_{2}\) in \(\mathbb{R}^{m}\) having no points in common for which \(d\left(E_{1}, E_{2}\right)=0\).

Show that a) the closure \(\bar{E}\) in \(\mathbb{R}^{m}\) of any set \(E \subset \mathbb{R}^{m}\) is a closed set in \(\mathbb{R}^{m}\); b) the set \(\partial E\) of boundary points of any set \(E \subset \mathbb{R}^{m}\) is a closed set; c) if \(G\) is an open set in \(\mathbb{R}^{m}\) and \(F\) is closed in \(\mathbb{R}^{m}\), then \(G \backslash F\) is open in \(\mathbb{R}^{m}\).

Show that if \(K_{1} \supset K_{2} \supset \cdots \supset K_{n} \supset \cdots\) is a sequence of nested nonempty compact sets, then \(\bigcap_{i=1}^{\infty} K_{i} \neq \varnothing\).