Let PR(X) denote the hyperspace of nonempty finite subsets of a topological space X with the Pixley– Roy topology. In this paper, motivated by [4], we introduced c f-covers and rc f-covers of X to establish the R-selective separability and the M-selective separability in PR(X) under the Pixley–Roy topology. We proved that the following statements are equivalent for a space X: (1) PR(X) is R-separable (resp., M-separable); (2) X satisfies S 1 (C rc f , C rc f) (resp., S fin (C rc f , C rc f)); (3) X is countable and each co-finite subset of X satisfies S 1 (C c f , C c f) (resp., S fin (C c f , C c f)); (4) X is countable and PR(X) has countable strong fan tightness (resp., PR(X) has countable fan tightness)