Exercise 3, § 2

a) Let $X$ be a compact metric space. If the closure in $X$ of every open ball is the closed ball of the same centre and radius, then every ball in $X$ is a connected set [if $x$ and $y$ are two points of $X$, and if $S$ is the closed ball with centre $x$ and radius $d(x, y)$, show that for each $\varepsilon > 0$ the set $A_{y, \varepsilon}$ of points of $S$ which can be joined to $y$ by a $V_\varepsilon$-chain contained in $S$ (Chapter II, § 4, no. 4) contains points $z$ such that $d(x, z) < d(x, y)$]. Deduce that $x \in A_{y, \varepsilon}$ for all $\varepsilon > 0$, arguing by contradiction and using Weierstrass’s theorem (Chapter IV, § 6, no. 1, Theorem 1).

b) In the space $Y = \mathbf{R}^2$ with the metric

let $X$ be the compact subspace consisting of all points $(x_1, x_2)$ such that either $x_1 = 0$ and $0 \leq x_2 \leq 1$, or $0 \leq x_1 \leq 1$ and $x_2 = 0$. Show that every ball in $X$ is connected but that the closure of an open ball is not necessarily the closed ball with the same centre and radius.