Linear topologies and sequential compactness in topological modules

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.