Homotopy Properties of the Space If(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 .