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?



subsets of groups which have to be closed no matter what
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](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6243&option_lang=eng) 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. 


