Geometrical properties of space $I_f(X)$ of idempotent probability measures

In the paper we proved that for a compact $X$ inclusion $I_f(X)\in ANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor $I_f$ preserves property of a compact to be $Q$-manyfold or a Hilbert cube, preserves property of map layers to be $ANR$-compact, compact $Q$-manyfold and a Hilbert cube (the finite sum of Hilbert cubes)