Grzegorczyk points and filters in boolean contact algebras

The purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from a philosophical interpretation of the results from the paper.