Why is it necessary that when we are defining a filtration of algebraic objects there must be a topology associated to the filtration?
For example, a descending filtration of group has the topology whose basis is the set of all translates of subgroups appearing in the filtration. Then for modules and ring there is I-adic topology. And so for other cases.
My question is-
Why is it necessary to have a topology associated to the filtration?
Does it come by default?
Please explain it.