On separability of the functional space with the open-point and bi-point-open topologies

Abstract

We study the property of separability of functional space C(X) with the open-point and bi-point-open topologies and show that it is consistent with ZFC that there is a set of reals of cardinality c such that a set C(X) with the open-point topology is not a separable space. We also show in a set model (the iterated perfect set model) that for every set of reals X, C(X) with the bi-point-open topology is a separable space. © 2016, Akadémiai Kiadó, Budapest, Hungary.