An uncountable Mittag-Leffler condition with an application to non-archimedean locally convex vector spaces

Mittag-Leffler condition ensures the exactness of the inverse limit of a family of short exact sequences indexed on a partially ordered set admitting a countable cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an ultrametric analogous of a result of V. P. Palamodov in relation with the acyclicity of Fréchet spaces with respect to the completion functor.