ИСТИНА

There are two covering dimensions, $\dim$ in the sense of \v Cech ($\dim X\le n$ if any finite open cover of $X$ has a finite open refinement of order $\le n$) and $\dim_0$ in the sense of Kat\v etov ($\dim_0 X\le n$ if any finite cozero cover of $X$ has a finite cozero refinement of order $\le n$). It is proved that the covering dimension of the Sorgenfrey plane $S\times S$ is infinite, while, as is well known, $\dim_0 S^\kappa=0$ for any cardinal $\kappa$. Examples of topological groups with similar properties are constructed, including a separable precompact Boolean group $G$ with linear topology (generated by open subgroups) with $\dim_0 G^\kappa=0$ and $\dim G=\infty$. The open problem of the existence in ZFC of topological groups whose underlying space is an $F$-space (i.e., a space in which any two disjoint cozero sets are functionally separated) not being $P$-spaces is touched on. It is proved that the existence of an Abelian $F$-group $G$ with $\dim_0 G<\infty$ and $\psi(G)\le \omega$ is equivalent to the existence of a Boolean group with the same properties and that the existence of an Abelian $F'$-group (or of an extremally disconnected group) $G$ with linear topology which is not a $P$-space implies the existence of a group of cardinality $\le 2^\omega$ with the same properties. \begin{question} Is it true that $ \dim_0 X\le \dim X$ for any completely regular space~$X$? \end{question}