Integrally closed algebras of analytic elements

Let K be an algebraically closed complete ultrametric field and let D be an infraconnected set such that the set H (D) of the Analytic Elements on D is a K-sub-algebra of K^D. If D has no T-filter we prove that H (D) is an integrally closed algebra even when D is not open (so that H (D) is not noetherian). Besides , we construct an infraconnected set with no T-filter and with empty interior which provides a typical illustration of this Theorem.