Spaces having a weakly-infinite-dimensional compactification (Q1072152)

The author introduces the property, small-weakly-infinite-dimensional (small-w.i.d.). He proves that for a separable metric space X, the following are equivalent, (1) X is small-w.i.d., (2) X has a small-w.i.d. completion, (3) X has a w.i.d. metric compactification. He shows that every complete, separable metric, totally disconnected space is small-w.i.d. Since it is known that there exist complete, separable metric, totally disconnected spaces that are infinite dimensional but not countable dimensional, then the preceding shows that there exists a metric compactum which is neither countable dimensional nor strongly infinite dimensional. This is of course the result due to \textit{R. Pol} [Proc. Am. Math. Soc. 82, 634-636 (1981; Zbl 0469.54014)].