On {C}ech-Completeness of the Space of τ-Smooth Idempotent Probability Measures

For the set of probability measures I(X), where X is a compact Hausdorff space, we propose a new way to introduce the topology by using 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 {C}ech-complete if and only if the given space X is {C}ech-complete.