On generalization of the $\mathcal{T}_{A_{I}}-$density topology

We present a further generalization of the $\mathcal{T}_{A_{I}}$-density topology introduced in (Wojciech Wojdowski, A category analogue of the generalization of Lebesgue density topology, Tatra Mt. Math. Publ., 42 (2009), 11-25 ) as a generalization of the $I-$density topology. We construct an ascending sequence $\{\mathcal{T}{_{A_{I(n)}}\}}_{n\in \mathbb{N% }}$ of density topologies which leads to the $\mathcal{T}_{A_{I(\omega )}}$% -density topology including all previous topologies. We examine several basic properties of the topologies.