Metrization of Idempotent Convex Compacta

Using a convenient subbase on the second hyperspace of a compactum with the Vietoris topology, we prove that the mapping that takes each closed non-empty subset $A$ of an $I$-convex compactum $X$ to its closed idempotent convex hull is continuous. This implies that each neighborhood of the diagonal $\Delta_X\subset X\times X$ contains an idempotent convex neighborhood. The main result is the theorem that the topology on an idempotent convex compactum $X$ is determined by a family of idempotent convex pseudometrics (with one idempotent convex metric if $X$ is metrizable).