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

of an

I I

-convex compactum

X X

to its closed idempotent convex hull is continuous. This implies that each neighborhood of the diagonal

\Delta_X\subset X\times X Δ X ⊂ X × X

contains an idempotent convex neighborhood. The main result is the theorem that the topology on an idempotent convex compactum

X X

is determined by a family of idempotent convex pseudometrics (with one idempotent convex metric if

X X

is metrizable).