Abstract
The extremal topology on an arbitrary set X was de ned as a maximal non-discrete topology [1] and [2]. In this paper we introduce an extremal topology on a fuzzy set X, which is also a maximal non-discrete topology, and it has to be in a speci c form. This form depends on some ultra lters = . We consider some properties for this kind of topologies when = is free. The subspaces and the base of this topology is also considered.