Exercise 4, § 2

A metric space $X$ is said to be an ultrametric space if its metric $d$ satisfies the inequality

for all $x, y, z$ in $X$ (which implies the triangle inequality) (cf. § 6, Exercise 2).

a) If $d(x, z) \neq d(y, z)$, show that $d(x, y) = \sup(d(x, z), d(y, z))$.

b) Let $V_r(x)$ be the open ball with centre $x$ and radius $r$. Show that $V_r(x)$ is both open and closed in $X$ (and hence that $X$ is totally disconnected) and that for each $y \in V_r(x)$ we have $V_r(y) = V_r(x)$.

c) Show that the closed ball $W_r(x)$ with centre $x$ and radius $r$ is both open and closed in $X$, and that for each $y \in W_r(x)$ we have

The distinct open balls of radius $r$ contained in $W_r(x)$ form a partition of $W_r(x)$, and the distance between any pair of these balls is equal to $r$.

d) If two balls (open or closed) of $X$ intersect, then one is contained in the other.

e) A sequence $(x_n)$ of points of $X$ is a Cauchy sequence if and only if $d(x_n, x_{n+1})$ tends to 0 as $n$ tends to infinity.

f) If $X$ is compact, show that for each $x_0 \in X$ the set of values of $d(x_0, x)$ in $X$ is a (finite or infinite) countable subset of $[0, +\infty]$, of which all the points, with the possible exception of 0, are isolated [for each value $r$ taken by $d(x_0, x)$, consider the least upper bound of $d(x_0, x)$ on the set of points where $d(x_0, x) < r$, and its greatest lower bound on the set of points where $d(x_0, x) > r$].