**Topology Induced by Filtration:**

Let be a descending filtration of a group such that for all Consider the collection of all cosets of subgroups in the filtration together with the empty set; we will denote this collection by The fact that for subgroups that is linearly ordered (which, together with the first fact, gives (where without loss of generality ) and ), and that is maximal in the collection of cosets (which, with Zorn’s lemma, gives closure under abritrary unions) makes a topology on . In particular, group multiplication can be shown to be continuous with respect to this topology–making a topological group. We will call this the **filtrated topology (or topology induced by filtration) of** by

Hence (linearly ordered) filtrations on a structure of “at least group-type” induce a topology. We can also induce gradings from filtrations of groups with the assumption that elements of the filtration are normal in We then define

in the same way we previously did for ideals in a ring. As a partition, a grading can also induce a topology, where the open sets are generated by elements of the partition. For example, in the grading of a group, let open sets be terms in the sum. Note that unions of terms are terms, the empty set can trivially be considered a term, and finite intersections of terms are terms.

In the case of an ideal of a ring the topology induced by the filtration

is called the **Krull topology** (or –**adic topology**) of by

**Spectral/Zariski Topology**:

Let be the set of prime ideals of a unital ring If is an ideal of let

Note that and Hence the ‘s form a basis of closed sets of The corresponding topology is called the **spectral (or Zariski) topology** of which we also denote by

[1] Bourbaki, N. *Commutative Algebra, Ch 1-7*. Springer-Verlag. 1989.