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In this paper, the core convex topology on a real vector space $X$, which isconstructed just by $X$ operators, is investigated. This topology, denoted by$\tau_c$, is the strongest topology which makes $X$ into a locally convexspace. It is shown that some algebraic notions $(closure ~ and ~ interior)$existing in the literature come from this topology. In fact, it is proved thatalgebraic interior and vectorial closure notions, considered in the literatureas replacements of topological interior and topological closure, respectively,in vector spaces not necessarily equipped with a topology, are actually nothingelse than the interior and closure with the respect to the core convextopology. We reconstruct the core convex topology using an appropriatetopological basis which enables us to characterize its open sets.  Furthermore, it is proved that $(X,\tau_c)$ is not metrizable when X isinfinite-dimensional, and also it enjoys the Hine-Borel property. Using theseproperties, $\tau_c$-compact sets are characterized and a characterization offinite-dimensionality is provided. Finally, it is shown that the properties ofthe core convex topology lead to directly extending various important resultsin convex analysis and vector optimization from topological vector spaces toreal vector spaces.

Reconstruction of the core convex topology and its applications in vector optimization and convex analysis