Selective separability and its variations

A space X is said to be selectively separable (=M-separable) if for each sequence { D n : n ∈ ω } of dense subsets of X, there are finite sets F n ⊂ D n ( n ∈ ω ) such that ⋃ { F n : n ∈ ω } is dense in X. On selective separability and its variations, we show the following: (1) Selective separability, R-separability and GN-separability are preserved under finite unions; (2) Assuming CH (the continuum hypothesis), there is a countable regular maximal R-separable space X such that X 2 is not selectively separable; (3) { 0 , 1 } c has a selectively separable, countable and dense subset S such that the group generated by S is not selectively separable. These answer some questions posed in Bella et al. (2008) [7].