  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)
