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Compatible topologies and bornologies on modules. (English) Zbl 0937.46046
The paper studies compatible topologies and bornologies in the context of modules. For a topological ring \(A\) with identity and a (unitary left) \(A\)-module \(E\), an \(A\)-module topology \(\tau \) on \(E\) and an \(A\)-module bornology \(\mathcal B\) on \(E\) are called compatible if \(\mathcal B\) is finer than the bornology of all \(\tau \)-bounded subsets of \(E\). In the first part of the paper, standard methods for obtaining new compatible topologies and bornologies from given ones are discussed (limits, products, and sums). In the second part, quasi-boundedness of certain topological modules of morphisms \((E,\tau ,\mathcal B)\to (E',\tau ',\mathcal B')\), where \(E\) and \(E'\) are two \(A\)-modules with compatible topologies and bornologies \((\tau ,\mathcal B)\) and \((\tau ',\mathcal B')\), respectively, is established.

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46A17 Bornologies and related structures; Mackey convergence, etc.
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