Finite dimensions defined by means of m-coverings

Abstract

We introduce and investigate finite dimensions$(m,n)\text{-}\text{tx}\mathrm{d}\mathrm{i}\mathrm{m}$ defined by means of $m$-coverings. Thesedimensions generalize the Lebesgue dimension: $\text{tx}\text{rmdim}=(2,1)\text{-}\text{tx}\mathrm{d}\mathrm{i}\mathrm{m}$. If $n<m$ and $(m,n)\text{-}\text{tx}\text{rmdim}X<ınfty$, then $X$ is weakly infinite-dimensional in thesense of Smirnov.