Homotopy Properties of the Space $I_f(X)$ of Idempotent Probability Measures

A subspace $I_f(X)$ of the space of idempotent probability measures on a given compact space $X$ is constructed. It is proved that if the initial compact space $X$ is contractible, then $I_f(X)$ is a contractible compact space as well. It is shown that the shapes of the compact spaces $X$ and $I_f(X)$ are equal. It is also proved that, given a compact space $X$, the compact space $I_f(X)$ is an absolute neighborhood retract if and only if so is $X$.