Non-Archimedean absolute value ($|\cdot |$)

A non-Archimedean absolute value satisfies the ultrametric inequality $|x+y|\le \max (|x|,|y|)$.

The ultrametric inequality is $d(x,z)\le \max (d(x,y),d(y,z))$. For unequal $|x|$ and $|y|$, it implies $|x+y|=\max (|x|,|y|)$.

An ultrametric space is a metric space satisfying the ultrametric inequality. Its open balls are clopen, and any two balls are either disjoint or one contains the other.

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