On condition for density-type topologies generated by functions

We discuss algebraic properties of density-type topologies generated by functions. We examine Δ 2 condition, similar to the condition considered in the theory of Orlicz spaces. We show that a topology T f coarser than the density topology is invariant under multiplication by nonzero numbers if and only if f fulfills Δ 2 condition. We also show that for other topologies generated by functions, connections between algebraic properties of T f and Δ 2 condition are more complicated.