Tomasz Filipczak

On monotone hull operations

We extend results of Elekes and Mathe on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ-ideals on the plane.

On homeomorphisms of density type topologies generated by functions

Abstract The paper is concerned with homeomorphisms on topological spaces (ℝ, Tƒ), where Tƒ is a generalization of the density topology generated by a function ƒ. It is shown that the density topology is not homeomorphic with any other topology Tƒ and that, under some assumptions, homeomorphic ƒ-density topologies have to be comparable.

Monotonicity theorems for the I-proximal local system

In paper [1] it was proved that if a local system S fulfils the so-called condition (SD), then the monotonicity of a function / is implied by some conditions of (S)-limits and (S)-derivatives of / . In our paper we show that the local system which consists of complements of sets I-sparse at a point fulfils the condition (SD). We also formulate some monotonicity theorems which follow from this fact.