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 of X x 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).