Strict density topology of the plane. category case

We study the properties of category density topology of the plane generated by a restricted convergence in the category of double sequences of characteristic functions and more interesting topology generated by a strict convergence in the category of the same sequences, which is a natural modification of a previous one. Similar problems for measure density were considered in [M. Filipczak, W. Wilczy´nski: Strict density topology on the plane. Measure case (in preparation)].