On extension of functions from countable subspaces

Three intermediate class of spaces $\mathscr{R}_1\subset \mathscr{R}_2\subset \mathscr{R}_3$ between the classes of $F$- and $\beta\omega$-spaces are considered. The $\mathscr{R}_1$- and $\mathscr{R}_3$-spaces are characterized in terms of the extension of functions. It is proved that the classes of $\mathscr{R}_1$-, $\mathscr{R}_2$-, $\mathscr{R}_3$-, and $\beta\omega$-spaces are not preserved by the Stone–Čech compactification.