  We establish some geometrical properties of the space of idempotent
probability measures. In particular, for a compact $X$ it is established that
if the space $I_{3}(X)\backslash X$ is hereditary normally, then $X$ is
metrizable; some subsets allocate of the space of idempotent probability
measures which are, respectively, $Z$-sets, $\max$-$\mbox{plus}$-convex
subsets, $G_\delta$-sets.
