Selective separability properties of Fréchet-Urysohn spaces and their products

Abstract

In this paper we study the behaviour of selective separability properties in the class of Frechét-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frechét-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\mathfrak{p}=\mathfrak{c}$, we construct such an example which is also zero-dimensional and ${\alpha }_4$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frechét-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two $H$-separable spaces is $mH$-separable.

31 pages; comments are welcome