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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.

##### MSC:
 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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