{{distinguish|Linear bus topology}}
In algebra, a '''linear topology''' on a left <math>A</math>-module <math>M</math> is a topology on <math>M</math> that is invariant under translations and admits a fundamental system of neighborhoods of <math>0</math> that consist of submodules of <math>M.</math><ref name="Lefschetz" /> If there is such a topology, <math>M</math> is said to be '''linearly topologized'''. If <math>A</math> is given a discrete topology, then <math>M</math> becomes a topological <math>A</math>-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."<ref name="exact" />
The term "linear topology" goes back to Lefschetz's work.<ref name="Lefschetz">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]</ref><ref name="exact">{{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 }}</ref>
== Examples and non-examples == *For each prime number ''p'', <math>\mathbb{Z}</math> is linearly topologized by the fundamental system of neighborhoods <math>0 \in \cdots \subset p^2 \mathbb{Z} \subset p\mathbb{Z} \subset \mathbb{Z}</math>. *Topological vector spaces appearing in functional analysis are typically not linearly topologized (since subspaces do not form a neighborhood system).
==See also== {{div col}} * {{annotated link|Ordered topological vector space}} * {{annotated link|Ring of restricted power series}} * {{annotated link|Topological abelian group}} * {{annotated link|Topological field}} * {{annotated link|Topological group}} * {{annotated link|Topological module}} * {{annotated link|Topological ring}} * {{annotated link|Topological semigroup}} * {{annotated link|Topological vector space}} {{div col end}}
==References== {{reflist}} * Bourbaki, N. (1972). Commutative algebra (Vol. 8). Hermann.
Category:Topological algebra Category:Topological groups
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