Selective separability properties of Fréchet–Urysohn spaces and their products
Tom 263 / 2023
Streszczenie
We study the behaviour of selective separability properties in the class of Fréchet–Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Fréchet–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 Fréchet–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.