On the 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.