A GENERALIZATION OF M-SEPARABILITY BY NETWORKS

All spaces are assumed to be Tychonoff. A space is M-separable if for every sequence (Dn : n ∈ ω) of dense subsets of X one can pick finite Fn ⊂ Dn, n ∈ ω, such that U n∈ω Fn is dense in X. Every space having a countable base is M-separable but not every space with countable network weight is M-separable. We introduce a new Menger type property defined by networks, called M-nw-selective property, such that every M-nw-selective space has countable network weight and is M-separable. By analogy, we also introduce H- and R- nw-selective spaces for Hurewicz and Rothberger type properties. Several properties of the new classes of spaces are studied and some questions are posed.