TITLE: subsets of groups which have to be closed no matter what
QUESTION [12 upvotes]: One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?

REPLY [25 votes]: Subsets of a group that are closed with respect to any Hausdorff group topology
are called unconditionally closed. 
Clearly, all algebraic sets are unconditionally closed, where a subset of a group $G$ is called algebraic if it is an intersection of finite unions of the sets of solutions to some equations with coefficient from $G$.  
A.A.Markov proved that for countable groups the converse is also true:
$$
\hbox{unconditionally closed = algebraic}.
$$
For uncountable groups, this is not always the case as follows from works of S. Shelah (under CH) and G. Hesse.