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 of that consists of submodules of [1] 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, "[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]
See also[edit]
- Ordered topological vector space
- Ring of restricted power series – Formal power series with coefficients tending to 0
- Topological abelian group – concept in mathematics
- Topological field – Algebraic structure with addition, multiplication, and division
- Topological group – Group that is a topological space with continuous group action
- Topological module
- Topological ring – ring where ring operations are continuous
- Topological semigroup – semigroup with continuous operation
- Topological vector space – Vector space with a notion of nearness
References[edit]
- ^ a b Ch II, Definition 25.1. in Solomon Lefschetz, Algebraic Topology
- ^ a b 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.
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