Linear topology

In algebra, a linear topology on a left -module is a topology on that is invariant under translations and admits a fundamental [system of neighborhood]s of that consist of submodules of If there is such a topology, is said to be linearly topologized. If is given a discrete topology, then becomes a topological -module with respect to a linear topology.
The notion is used more commonly in algebra than in analysis. Indeed, "opological vector spaces with linear topology form a natural class of topological vector spaces over discrete fields, analogous to the class of locally convex topological vector spaces over the normed fields of real or complex numbers in functional analysis."
The term "linear topology" goes back to Lefschetz' work.

Examples and non-examples

For each prime number p, is linearly topologized by the fundamental system of neighborhoods.
Topological vector spaces appearing in functional analysis are typically not linearly topologized.