linear topology

{{distinguish|Linear bus topology}}

In algebra, a linear topology on a left $A$-module $M$ is a topology on $M$ that is invariant under translations and admits a fundamental system of neighborhoods of $0$ that consist of submodules of $M.$ If there is such a topology, $M$ is said to be linearly topologized. If $A$ is given a discrete topology, then $M$ becomes a topological $A$-module with respect to a linear topology.

The notion is used more commonly in algebra than in analysis. Indeed, "[t]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.Ch II, Definition 25.1. in Solomon Lefschetz, [https://books.google.com/books/about/Algebraic_Topology.html?id=JzhPAQAAIAAJ&source=kp_book_description Algebraic Topology]{{cite journal |last1=Positselski |first1=Leonid |title=Exact categories of topological vector spaces with linear topology |journal= Moscow Mathematical Journal|date=2024 |volume=24 |issue=2 |pages=219–286|doi=10.17323/1609-4514-2024-24-2-219-286 |arxiv=2012.15431 }}