On some recent selective properties involving networks

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 propert