# Linear topology

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Linear_topology
> Markdown URL: https://mediated.wiki/source/Linear_topology.md
> Source: https://en.wikipedia.org/wiki/Linear_topology
> Source revision: 1353669627
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{distinguish|Linear bus topology}}

In [algebra](/source/algebra), a '''linear topology''' on a left <math>A</math>-module <math>M</math> is a [topology](/source/topology) on <math>M</math> that is invariant under translations and admits a [fundamental system of neighborhood](/source/fundamental_system_of_neighborhood)s 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](/source/Topological_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](/source/functional_analysis)."<ref name="exact" />

The term "linear topology" goes back to [Lefschetz](/source/Solomon_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.](/source/Nicolas_Bourbaki) (1972). Commutative algebra (Vol. 8). Hermann.

Category:Topological algebra
Category:Topological groups

{{algebra-stub}}

---
Adapted from the Wikipedia article [Linear topology](https://en.wikipedia.org/wiki/Linear_topology) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Linear_topology?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.