On the Čech-Completeness of the Space of τ-Smooth Idempotent Probability Measures

For the set I(X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I(X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space Iτ(X) of τ-smooth idempotent probability measures on X and show that the space Iτ(X) is Čech-complete if and only if the given space X is Čech-complete.