On \v{C}ech-Completeness of the Space of $\tau$-Smooth Idempotent Probability Measures

For the set of probability measures ${\sf I}(X)$, 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 ${\sf I}(X)$ is also a compact Hausdorff space. For a Tychonoff space $X$, we consider the topological space ${\sf I_{\tau}}(X)$ of $\tau$-smooth idempotent probability measures on $X$,and show that the space ${\sf I_{\tau}}(X)$ is \v{C}ech-complete if and only if the given space $X$ is \v{C}ech-complete.

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