Lower CS-Closed Sets and Functions

In this work, we introduce a class of convex sets, LCSCF X., of a locally convex separated and not necessarily separable topological vector space X. They are called the lower CS-closed sets. This class contains the CS-closed sets, satisfies the property core C.sint C., ;CgLCSCF X.when X is metrizable barrelled, and is stable under many operations. Among them, the projection and the denumerable intersection. We characterize the lower CS-closed functions i.e., the functions who have a lower CS-closed epigraph. as marginal functions of CS-closed ones and show that they are very stable too. We establish an open mapping and a closed graph theorem for the lower CS-closed relations. Finally, we show that every real extended valued lower CS-closed function defined on a metrizable barrelled space is continuous on the interior of its domain. This result allows us to extend classical theorems of convex duality by replacing lower semicontinuous functions by lower CS-closed ones. More than that, it systematizes and extends some methods of convex analysis.