On ($O$-$C$)-compactness of the space of idempotent probability measures

Abstract

We establish a connection between connected sets and max-plus connected sets. Then this connection is applied between spaces of ordinary probability measures and idempotent (max-plus) probability measures on a given Tychonoff space. We show that for every Tychonoff space $X$ the space $I_{\beta}(X)$ is max-plus convex and therefore, path max-plus-connected. Combining these properties with topological arguments we prove that for a Tychonoff space $X$ the space $I_{\beta}(X)$ is ($O$-$C$)-compact. Auxiliary propositions relating path connectedness and max-plus connectedness are established and used in the derivation of the main results.