This is always true. Every open ball in an <ultrametric space> is also closed: a point outside a ball has a disjoint ball of the same radius around it. Distinct points can therefore be separated by clopen sets, so every connected subset is a singleton and $K$ is <totally disconnected space>[totally disconnected].

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 1 / b / iv / Solution

{wiki=Totally_disconnected_space}

A topological space is totally disconnected when its only connected subsets are singletons.


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