On some topologies of O'Malley's type on the plane (Q1803980)

The paper consists of two parts. In the first one can find several theorems and examples showing the connections (or their lack) between measurability, \(d\times d\)-quasi-continuity and \(d\times d\)-cliquishness of functions of two variables and \(d\)-quasi-continuity (or cliquishness) of its sections. Here \(d\) denotes the density topology. As an example let us quote: there exists a measurable function which is not \(d\times d\)- cliquish. The second part is devoted to similar questions with the topology \(d\) replaced by a new topology \(q-a\) Hashimoto topology generated by a sigma ideal of meager sets. Here is given an example: every \(q\times q\)-cliquish function has the Baire property.