We introduce and investigate new selection principles involving countable networks in cosmic tychonoff spaces, namely, M-nw-selective, R-nw-selective, and H-nw-selective. These spaces represent a strengthening of both M-separability, R-separability, and H-separability, as well as the Menger, Rothberger, and Hurewicz properties.

We also define and investigate two new games: the R-nw-selective game and the M-nw-selective game, which arise naturally from their corresponding selection principles.

We give consistent results, and we define trivial R-, H-, and M-nw-selective spaces the cosmic ones having cardinality and weight strictly less than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$, respectively.

We establish that spaces with cardinalities greater than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ fail to possess the R-, H-, and M-nw-selective properties, respectively.

Non-trivial examples, therefore, 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. Finally, we study relations between nw-selective properties and a strong version of the HFD property.

A joint work with Leandro Aurichi, Maddalena Bonanzinga, Santi Spadaro, Lyubomyr Zdomskyy