  An ultrametric preserving function $f$ is said to be strongly ultrametric
preserving if ultrametrics $d$ and $f \circ d$ define the same topology on $X$
for each ultrametric space $(X,d)$. The set of all strongly ultrametric
preserving functions is characterized by several distinctive features. In
particular, it is shown that an ultrametric preserving $f$ belongs to this set
iff $f$ preserves the property to be compact.
