It is common for mathematicians to try and merge different structures which they find interesting. Topological groups and Topological rings are examples of this, where one defines a topology and a group (resp. ring) structure on the same underlying set, and requires that the group operations are continuous with respect to this tolology.
There are many different ways to define a Topology on a given group to achieve this, but if you take some collection of normal subgroups of a given group, and all their cosets as a subbase, the resulting topology produces a topological group. If you take a collection of ideals of a given ring, and all their 'cosets' as a subbase, the resulting topology produces a Topological ring.
In both these examples, an arbitrary collection of kernels of homomorphisms could be used to define a Topological structure that works nicely with the algebraic structure. Is this a coincidence? Does this work when trying to define topologies on other kinds of algebraic structure where the notion of a kernel makes sense? Is there some sort of underlying reason for this?