Dugundji Compacta and the Space of Idempotent Probability Measures

Abstract

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.