Abstract:
Three intermediate classes $\mathscr R_1\subset\mathscr R_2\subset\mathscr R_3$ between the classes of $F$-spaces and of $\beta\omega$-spaces are considered. It is proved that products of infinite $\mathscr R_2$-spaces and, under the assumption of the existence of a discrete ultrafilter, of infinite $\beta\omega$-spaces are never homogeneous. Under additional set-theoretic assumptions, the metrizability of any compact subspace of a countable product of homogeneous $\beta\omega$-spaces is proved.
Keywords:$\mathscr R_1$-space, $\mathscr R_2$-space, $\mathscr R_3$-space, Rudin–Keisler order, Rudin–Blass order, $\beta\omega$-space, NNCPP$_\kappa$, homogeneity of products of topological spaces.
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