On Δ2 condition for density-type topologies generated by functions

We discuss algebraic properties of density-type topologies generated by functions. We examine Δ2 condition, similar to the condition considered in the theory of Orlicz spaces. We show that a topology Tf coarser than the density topology is invariant under multiplication by nonzero numbers if and only if f fulfills Δ2 condition. We also show that for other topologies generated by functions, connections between algebraic properties of Tf and Δ2 condition are more complicated.