topology on , said to be associated to the filtration. A basis for this topology is the set of all cosets of subgroups appearing in the filtration, that is, a subset of is defined to be open if it is a union of sets of the form , where and is a natural number. The topology associated to a filtration on a group makes into a topological group. The topology associated to a filtration on a group is Hausdorff if and only if . If two filtrations and are defined on a group , then the identity map from to , where the first copy of is given the -topology and the second the -topology, is continuous if and only if for any there is an such that , that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other. Rings and modules: descending filtrations Given a ring and an -module , a descending filtration of is a decreasing sequence of submodules . This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups. An important special case is known as the -adic topology (or -adic, etc.): Let be a commutative ring, and an ideal of . Given an -module , the sequence of submodules of forms a filtration of . The -adic topology on is then the topology associated to this filtration. If is just the ring itself, we have defined the -adic topology on . When is given the -adic topology, becomes a topological ring. If an -module is then given the -adic topology, it becomes a topological -module, relative to the topology given on . Rings and modules: ascending filtrations Given a ring and an -module , an ascending filtration of is an increasing sequence of submodules . In particular, if is a field, then an ascending filtration of the -vector space is an increasing sequence of vector subspaces of . Flags are one important class of such filtrations. Sets A maximal filtration of a set is equivalent to an ordering (a permutation) of the set. For instance, the filtration corresponds to the ordering . From the point