Dugundji Compacta and the Space of Idempotent Probability Measures

For a given group $$(G,X,\alpha)$$ of topological transformations on a Tikhonov space $$X$$ , a group $$(I(G, X), I(X), I(\alpha))$$ of topological transformations on the space $$I(X)$$ of idempotent probability measures is constructed. It is shown that, if the action $$\alpha$$ of the group $$G$$ is open, then the action $$I(\alpha)$$ of the group $$I(G,X)$$ is also open; while an example is given showing that the openness of the action $$\alpha$$ is substantial. It has been established that, if the diagonal product $$\Delta f_{p}$$ of a given family $$\{f_{p}, f_{pq}; A\}$$ of continuous mappings is an embedding, then the diagonal product $$\Delta I(f_{p})$$ of the family $$\{I(f_{p}), I(f_{pq}); A\}$$ of continuous mappings is also an embedding. A Dugundji compactness criterion for the space of idempotent probability measures is obtained.

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