Problem 11
Prove that \(\bar{A}\) is contained in any closed superset of \(A\) and is the intersection of all such supersets.
Problem 11
Prove that \(\bar{x}\) and \(\bar{y}\) are parallel iff $$ \frac{x_{1}}{y_{1}}=\frac{x_{2}}{y_{2}}=\cdots=\frac{x_{n}}{y_{n}}=c \quad\left(c \in E^{1}\right) $$ where " \(x_{k} / y_{k}=c "\) is to be replaced by " \(x_{k}=0\) " if \(y_{k}=0\).
Problem 11
Give examples to show that an infinite intersection of open sets may not be open, and an infinite union of closed sets may not be closed. [Hint: Show that $$ \bigcap_{n=1}^{\infty}\left(-\frac{1}{n}, \frac{1}{n}\right)=\\{0\\} $$ and $$ \left.\bigcup_{n=2}^{\infty}\left[\frac{1}{n}, 1-\frac{1}{n}\right]=(0,1) .\right] $$
Problem 14
Prove that \(A^{0}\), the interior of \(A\), is the union of all open globes contained in \(A\) (assume \(\left.A^{0} \neq \emptyset\right)\). Deduce that \(A^{0}\) is an open set, the largest contained in \(A .^{3}\)
