What you've described as a topology, is actually a family of analytic P-ideals on $P(\mathbb{N})$ called the "summable ideals." They are studied for different reasons, and the general search terms to use are "summable ideal" or more generally "analytic P-ideal".

What you've described as a topology, is actually a family of analytic P-ideals on $P(\mathbb{N})$ called the "summable ideals." They are studied for different reasons, and the general search terms to use are "summable ideal" or more generally "analytic P-ideal".