Searchable Sets , Dubuc - Penon Compactness

We show that a number of contenders for an abstract and general notion of compactness, applicable in particular to computability theory and constructive mathematics, coincide in some well known frameworks. We consider compactness of sets rather than of spaces, where we replace topologies by the restriction to constructive reasoning, as in the work by a number of authors, including Penon, Dubuc, Taylor and Escardó. Sets here are conceived in a very liberal way, including types of HA and Martin Löf type theory, and objects of toposes, among others. Some of the equivalences require instances of the axiom of choice, which are available in some of the above frameworks but not all, as is well known. We relate the instances of the axiom of choice applied in the above equivalences to the topological notion of total separatedness.