$\mathscr {R}$-compact spaces with $\operatorname {X} < \operatorname {Exp}_{\mathscr {R}} X$

Abstract:

Answering a question of Arhangel’skii, we show—under GCH—that for most cardinals $\mathfrak {m}$ there exists an $\mathscr {R}$-compact space $X$ such that $\operatorname {weight} X = \mathfrak {m}$ but $X$ does not embed in a closed fashion into the product of $\mathfrak {m}$ copies of $\mathscr {R}$.