On the Homogeneity of Products of Topological Spaces

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.