On Čech-completeness of the space of $\tau$-smooth idempotent measures

For a Tychonoff space $X$, we study the space $I_\tau (X)$ of idempotent probability $\tau$ -smooth measures on $X$. Some types of open and closed subsets of the space of idempotent probability measures are noted. In the set of idempotent probability measures, the base of the product topology is introduced and it is shown that for a compact Hausdorff space $X$ the topological space $I(X)$ is also a compact Hausdorff space. Then we establish that the space $I(X)$ of idempotent probability measures is Čech-complete if and only if the given Tychonoff space $X$ is Čech-complete.