Density topology involving microscopic sets and category

Abstract

Analogously to the method used for $c-$density point, we introduce the
notion of $c_{\mu }-$density point for arbitrary set $A$ having the Baire
property,  replacing the covergence in measure with the convergence with
respect to the $\sigma -$ideal of microscopic sets.

Then we consider the density type topology $\mathcal{T}_{c_{\mu }}$ generated by $c_{\mu }-$ density operator and study basic properties of $\mathcal{T}_{c_{\mu }}$ in
comparison with the classical density, $\mathcal{I}-$density and $c-$density
topologies.