  A group $G$ is called hereditarily non-topologizable if, for every $H\le G$,
no quotient of $H$ admits a non-discrete Hausdorff topology. We construct first
examples of infinite hereditarily non-topologizable groups. This allows us to
prove that c-compactness does not imply compactness for topological groups. We
also answer several other open questions about c-compact groups asked by
Dikranjan and Uspenskij. On the other hand, we suggest a method of constructing
topologizable groups based on generic properties in the space of marked
$k$-generated groups. As an application, we show that there exist non-discrete
quasi-cyclic groups of finite exponent; this answers a question of Morris and
Obraztsov.
