On some recent selective properties involving networks

Abstract In this paper, we investigate R-, H-, and M- nw -selective properties introduced in [Bonanzinga, M.—Giacopello, D.: A generalization of M-separability by networks , Atti Accad. Peloritana dei Pericolanti Cl. Sci. Fis. Mat. Nat. 101(2) (2023), A11]. In particular, we provide consistent uncountable examples of such spaces and we define trivial R-, H-, and M- nw -selective spaces the ones with countable net weight having, additionally, the cardinality and the weight strictly less then cov( 𝓜 ), 𝔟, and 𝔡, respectively. Since we establish that spaces having cardinalities more than cov( 𝓜 ), 𝔟, and 𝔡, fail to have the R-, H-, and M- nw -selective properties, respectively, non-trivial examples should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R- nw -selective and H- nw -selective spaces and we establish some limitations to constructions of non-trivial examples. Moreover, we consistently prove the existence of two H- nw -selective spaces whose product fails to be M- nw -selective. Finally, we study some relations between nw -selective properties and a strong version of the HFD property.