Weakly infinite-dimensional spaces

In this survey article two new classes of spaces are considered: $m$-$C$-spaces and $w$-$m$-$C$-spaces, $m=2,3,\dots,\infty$. They are intermediate between the class of weakly infinite-dimensional spaces in the Alexandroff sense and the class of $C$-spaces. The classes of $2$-$C$-spaces and $w$-$2$-$C$-spaces coincide with the class of weakly infinite-dimensional spaces, while the compact $\infty$-$C$-spaces are exactly the $C$-compact spaces of Haver. The main results of the theory of weakly infinite-dimensional spaces, including classification via transfinite Lebesgue dimensions and Luzin–Sierpińsky indices, extend to these new classes of spaces. Weak $m$-$C$-spaces are characterised by means of essential maps to Henderson's $m$-compacta. The existence of hereditarily $m$-strongly infinite-dimensional spaces is proved.