  In this paper, the core convex topology on a real vector space $X$, which is
constructed just by $X$ operators, is investigated. This topology, denoted by
$\tau_c$, is the strongest topology which makes $X$ into a locally convex
space. It is shown that some algebraic notions $(closure ~ and ~ interior)$
existing in the literature come from this topology. In fact, it is proved that
algebraic interior and vectorial closure notions, considered in the literature
as replacements of topological interior and topological closure, respectively,
in vector spaces not necessarily equipped with a topology, are actually nothing
else than the interior and closure with the respect to the core convex
topology. We reconstruct the core convex topology using an appropriate
topological basis which enables us to characterize its open sets.
  Furthermore, it is proved that $(X,\tau_c)$ is not metrizable when X is
infinite-dimensional, and also it enjoys the Hine-Borel property. Using these
properties, $\tau_c$-compact sets are characterized and a characterization of
finite-dimensionality is provided. Finally, it is shown that the properties of
the core convex topology lead to directly extending various important results
in convex analysis and vector optimization from topological vector spaces to
real vector spaces.
