| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
18 |
|
4.I.1E |
2002 |
(a) Let $(X, d)$ be a metric space containing the point $x_{0}$, and let
$$U=\left{x \in X: d\left(x, x_{0}\right)<1\right}, \quad K=\left{x \in X: d\left(x, x_{0}\right) \leqslant 1\right}$$
Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.
(b) Let $X$ be a normed space with norm $|\cdot|$, and let
$$U={x \in X:|x|<1}, \quad K={x \in X:|x| \leqslant 1}$$
Is $U$ necessarily the largest open subset of $K$ ? Is $K$ necessarily the smallest closed set that contains $U$ ? Justify your answers.