ИСТИНА

A subset $A$ of a group $G$ is unconditionally closed in $G$ if it is closed in each Hausdorff group topology on $G$. A natural example of unconditionally closed sets is the solution sets of equations in $G$, as well as their finite unions and arbitrary intersections. Such sets are said to be algebraic. A natural necessary condition for a group to admit a nondiscrete Hausdorff group topology is that the complement to the identity must not be algebraic in this group. In 1945, Markov asked whether this condition is also sufficient and the more general question of whether any unconditionally closed set is algebraic. This question has a negative answer; the uncountable Jonsson group constructed by Shelah provides a CH example. The first ``naive'' example was constructed by Hesse in his 1979 dissertation. The current state-of-the-art in the field is delineated. Some positive results are presented.