On Immediate Extensions of Ultrametric Spaces

The purpose of this paper is to define immediate extensions of ultrametric spaces with a totally ordered value set. It will be proved that an ultrametric space (X, d) with value set F is maximal if and only if it is pseudocomplete, i. e. any pseudoconvergent sequence of (X, d) has a pseudolimit in X. Every ultrametric space will be shown to possess a maximal immediate extension, but its uniqueness can only be obtained within the more specific class of essential extensions.