Π-completeness of the space of idempotent probability measures

For a Tychonoff space X, we consider the space Iβ(X) of idempotent probability measures with compact support. Using a cover γX of the origin space X we build an open cover γ0 of Iβ(X). Then we show that if γX is a disjoint system then γ0 is a also disjoint system. We obtain that the space of all idempotent probability measures on the Stone-Cech compactification ˇ of X is a perfect compactification of Iβ(X). Finally, applying the Π-completeness criterion we establish that Iβ(X) is Π-complete if and only if the given Tychonoff space X is Π-complete.