Modular topologies on vector spaces

This paper addresses the topological structures induced on vector spaces by convex modulars that do not satisfy the $\Delta_2$ condition, with particular focus on their applications to variable exponent spaces such as $\ell^{(p_n)}$ and $L^{p(\cdot)}$. The motivation behind this investigation is its applicability to the study of boundary value problems involving the variable exponent $p(x)$-Laplacian when $p(x)$ is unbounded, a line of research recently opened by the authors. Fundamental topological properties are analyzed, including separation axioms, countability axioms, and the relationship between modular convergence and classical topological concepts such as continuity. Attention is given to the relation between modular and norm topologies. Special emphasis is placed on the openness of modular balls, the impact of the $\Delta_2$-condition, and duality with respect to modular topologies.