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

TL;DR

This paper investigates the core convex topology on real vector spaces, showing its fundamental properties, how it relates to algebraic notions, and its applications in extending convex analysis and vector optimization.

Contribution

It reconstructs the core convex topology, characterizes its open sets, and demonstrates its role in extending convex analysis and vector optimization to general vector spaces.

Findings

The core convex topology is the strongest locally convex topology on X.

X, τ_c is not metrizable in infinite dimensions but has the Hine-Borel property.

Properties of τ_c enable extension of convex analysis results beyond topological vector spaces.

Abstract

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.