# Ultrametric Counterexample

It is known fact that every ultrametric space is a metric space but the converse need not be true. As the class of metric spaces is larger than that of ultrametric spaces, anybody give a counter example for "Result holds in metric space that is not true in ultrametric space".

• Do you not mean the other way around? Any fact that is true of all metric spaces will still be true of all ultrametric spaces... – Eric Wofsey Nov 17 '16 at 5:40

I assume that you really want a result true for all ultrametric spaces that is not true for all metric spaces. Perhaps the simplest is that every ultrametric space is zero-dimensional: it has a base of clopen sets. This is immediate from the fact that if $\langle X,d\rangle$ is ultrametric, each open ball $B(x,r)$ for $x\in X$ and $r>0$ is also closed. To see this, suppose that $y\notin B(x,r)$, so that $d(x,y)\ge r$. Then $B(y,r)\cap B(x,r)=\varnothing$, for if there were some $z\in B(x,r)\cap B(y,r)$, we’d have
$$r\le d(x,y)\le\max\{d(x,z),d(y,z)\}<r\;,$$
Of course $\Bbb R$ with the usual metric is not zero-dimensional, since every zero-dimensional space is totally disconnected, and $\Bbb R$ is connected.