An ultrametric space is a metric space whose distance function satisfies the strong triangle inequality. In an ultrametric space, many curious properties are valid. For example, every triangle is an isosceles with the unequal side (if any) being shortest. In this paper, we observe that many of the unusual properties in an ultrametric space are actually equivalent to the strong triangle inequality. On the other hand, a simple example shows that some properties valid in an ultrametric space, such as every open ball is closed, do not imply the strong triangle inequality.

strong-triangle inequality, ultrametric spaces.