The Kripke schema in metric topology

Kripke’s schema with parameters turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke’s schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics. In this paper we show that a certain version of Kripke’s schema with parameters is equivalent to either of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. By so doing we use Kripke’s schema as a point of reference for classifying theorems of classical mathematics within the informal variant of the constructive reverse mathematics programme put forward by Ishihara [7, 8].1 As for the latter, the overall framework of the present note is Bishop-style constructive mathematics [1, 2, 3, 4, 10], which can be thought of as mathematics carried out with intuitionistic logic [11]. The Kripke Schema can be stated as follows [3]: For each proposition P there is an increasing binary sequence (an) such that P holds if and only if an = 1 for some n. Clearly the Kripke schema follows from the law of excluded middle: set an = 1 (respectively, an = 0) for every n whenever P (respectively, :P ) holds. 1For more references and other authors on constructive reverse mathematics see [9].