On metrizable spaces in dimension theory

We present a small variation ofMrowka’s recent technique for producing metrizable spaces with non-coinciding dimensions. This variation has several uses. First, it is easier to $\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\theta $ many of the important properties of spaces constructed this way. Secondly, it is more general, allowing for each complete separable metric space $X $, a zero-dimensional and metrizable space space $M(X) $ with, consistently, the same covering dimension as $X $. As an application, we consistently produce, for each $n\in N $ , a zero-dimensional metrizable space $X_{n} $ satisfying $n=\dim X_{n}=\dim$ $(X_{n})^{\omega} $. Mathematics Subject Classification (1991): $54\mathrm{F}45