On Generalization of the 𝒯_AI-Density Topology

We present a further generalization of the 𝒯_AI-density topology introduced in [W. 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 {𝒯_AI(n)}_n∈ℕ of density topologies which leads to the 𝒯_AI(ω)-density topology including all previous topologies. We examine several basic properties of the topologies.