Linear topology

Not to be confused with Linear bus topology.

In algebra, a linear topology on a left ${\displaystyle A}$-module ${\displaystyle M}$ is a topology on ${\displaystyle M}$ that is invariant under translations and admits a fundamental system of neighborhood of ${\displaystyle 0}$ that consists of submodules of ${\displaystyle M.}$^[1] If there is such a topology, ${\displaystyle M}$ is said to be linearly topologized. If ${\displaystyle A}$ is given a discrete topology, then ${\displaystyle M}$ becomes a topological ${\displaystyle A}$-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."^[2]

The term "linear topology" goes back to Lefschetz' work.^[1][2]

Examples and non-examples

• For each prime number p, ${\displaystyle \mathbb {Z} }$ is linearly topologized by the fundamental system of neighborhoods ${\displaystyle 0\in \cdots \subset p^{2}\mathbb {Z} \subset p\mathbb {Z} \subset \mathbb {Z} }$.
• Topological vector spaces appearing in functional analysis are typically not linearly topologized (since subspaces do not form a neighborhood system).

See also

• Ordered topological vector space
• Ring of restricted power series – Formal power series with coefficients tending to 0Pages displaying short descriptions of redirect targets
• Topological abelian group – topological group whose group is abelianPages displaying wikidata descriptions as a fallback
• Topological field – Algebraic structure with addition, multiplication, and divisionPages displaying short descriptions of redirect targets
• Topological group – Group that is a topological space with continuous group action
• Topological module
• Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback
• Topological semigroup – semigroup with continuous operationPages displaying wikidata descriptions as a fallback
• Topological vector space – Vector space with a notion of nearness

References

1. Ch II, Definition 25.1. in Solomon Lefschetz, Algebraic Topology
2. Positselski, Leonid (2024). "Exact categories of topological vector spaces with linear topology". Moscow Math. Journal. 24 (2): 219–286.

• Bourbaki, N. (1972). Commutative algebra (Vol. 8). Hermann.