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Metrics with good Corona properties

In this note, we give several definitions of metric corona properties which could be of interest in Set Topology, Functional Analysis and Approximation Theory, and prove that there are complete metrizable t.v.s. which are nice in the sense that they have a metric which is invariant by translations, but they do not have good corona properties. All classical function spaces satisfy good corona properties but it is an open question to know if this also holds for the more general setting of locally convex t.v.s.