# Linearly-compact module

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A topological module over a topological ring that has a basis (cf. Base) of neighbourhoods of zero consisting of submodules, and in which every centred system (or filter base, cf. also Centred family of sets) consisting of cosets with respect to closed submodules has a non-empty intersection. Every linearly-compact module is a complete topological group.

A linearly-compact module is called a linearly-compact module in the narrow sense if every continuous homomorphism onto a topological module that has a basis of neighbourhoods of zero consisting of submodules, is open (cf. Open mapping). A topological module is a linearly-compact module in the narrow sense if and only if it is a complete topological group and if every quotient module of it with respect to an open submodule is an Artinian module. In particular, an Artinian module in the discrete topology is a linearly-compact module in the narrow sense. Thus, linearly-compact modules in the narrow sense are the topological analogues of Artinian modules.

Direct products, closed submodules, quotient modules with respect to closed submodules, and continuous homomorphic images having a basis of neighbourhoods of zero consisting of submodules of linearly-compact modules (in the narrow sense) are themselves linearly-compact modules (in the narrow sense).

#### References

 [1] S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1955) [2] D. Zelinsky, "Linearly compact modules and rings" Amer. J. Math. , 75 : 1 (1953) pp. 79–90 [3a] H. Leptin, "Linear kompakte Moduln und Ringe" Math. Z. , 62 (1955) pp. 241–267 [3b] H. Leptin, "Linear kompakte Moduln und Ringe II" Math. Z. , 66 (1957) pp. 289–327