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Abstract
We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring whose invertibles approach 0 has the trivial topology. We also show that every sequentially compact set in a topological module is bounded if the module is over an absolute semi-valued ring whose set of invertibles is separable and its closure contains 0. Finally, we find sufficient conditions for a sequentially compact neighborhood of 0 to force the corresponding module to be finitely generated.
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