91Ó°ÊÓ

Show that the union of a compact set and a finite set is compact.

(a) Show that the intersection of any number of compact sets is compact. (b) Show that the union of any finite number of compact sets is compact. (c) Show that the union of an infinite number of compact sets may not be compact.

Given \(\mathbb{R}^{2}\) with the metric \(d(x, y)=\left|y_{1}-x_{1}\right|+\left|y_{2}-x_{2}\right|, x=\left(x_{1}, x_{2}\right), y=\left(y_{1}, y_{2}\right)\). Describe (and sketch) the ball with center at \((0,0)\) and radius 1 .

Given \(\mathbb{R}^{1}\) with \(d(x, y)=|x-y|\). Show that a finite set consists only of isolated points. Is it true that a set consisting only of isolated points must be finite?

Show that any family of pairwise disjoint open intervals in \(\mathbb{R}^{1}\) is countable. [Hint: Set up a one-to-one correspondence between the disjoint intervals and a subset of the rational numbers. (Theorem \(6.15(\mathrm{a})\).) ]

Let \(\mathscr{A}\) be the space of sequences \(x=\left\\{x_{1}, x_{2}, \ldots, x_{n}, \ldots\right\\}\) in which only a finite number of the \(x_{i}\) are different from zero. In \(\mathscr{A}\) define \(d(x, y)\) by the formula $$ d(\boldsymbol{x}, \boldsymbol{y})=\max _{1 \in i<\infty}\left|x_{i}-y_{i}\right| $$ (a) Show that \(\mathscr{A}\) is a metric space. (b) Find a closed bounded set in \(\mathscr{A}\) which is not compact.

Let \(G\) be an open set in \(\mathbb{R}^{1}\). Show that \(G\) can be represented as the union of open intervals with rational endpoints.

Let \(B\) be the set of points in \(R^{2}\) both of whose coordinates are rational. Show that \(B\) is not connected.

. Let \(A\) and \(B\) be connected sets in a metric space with \(A-B\) not connected and suppose \(A-B=C_{1} \cup C_{2}\) where \(\bar{C}_{1} \cap C_{2}=C_{1} \cap \bar{C}_{2}=\varnothing\). Show that \(B \cup C_{1}\) is connected.

Suppose \(f: A \rightarrow \mathbb{R}^{1}\) is continuous on \(A\), a compact set in a metric space. Show that the range of \(f\) contains its supremum and infimum (Theorem 6.30).