Let PR(X) denote the hyperspace of nonempty finite subsets of a topological space X with Pixley– Roy topology. In this paper, by introducing closed-miss-finite networks and using principle ultrafilters, we proved that the following statements are equivalent for a space X: (1) PR(X) is weakly Rothberger; (2) X satisfies S 1 (Π rc f , Π wrc f); (3) X is separable and X − {x} satisfies S 1 (Π c f , Π wc f) for each x ∈ X; (4) X is separable and each principal ultrafilter F [x] in PR(X) is weakly Rothberger in PR(X). We also characterize the weakly Menger property and the weakly Hurewicz property of PR(X).