A locally profinite group is a Hausdorff, locally compact, totally disconnected topological group. It has a neighbourhood basis at the identity consisting of compact-open subgroups. In smooth representation theory this is also called an -group; that terminology is unrelated to a lattice-ordered group.
A compact-open subgroup is both a compact subset and an open subgroup. Its coset space is discrete, and compact subsets meet only finitely many cosets. Finite intersections of compact-open subgroups are again compact-open, making them useful common stabilizers for finitely many vectors.
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A **locally profinite group** is a type of group that is constructed from profinite groups, which are groups that are isomorphic to an inverse limit of finite groups. Formally, a locally profinite group can be defined as a group \( G \) that has a neighborhood basis at the identity consisting of open subgroups that are profinite.