On homeomorphisms of density type topologies generated by functions

Abstract

The paper is concerned with homeomorphisms on topological spaces $\left(\mathbb{R},\mathcal{T}_{f}\right) $, where $\mathcal{T}_{f}$ is a generalization of the density topology generated by a function $f$. It is
shown that the density topology is not homeomorphic with any other topology $
\mathcal{T}_{f}$ and that, under some assumptions, homeomorphic $f$-density
topologies have to be comparable.