Geometrical properties of the space of idempotent probability measures

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.