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Ideal Banach category theorems and functions

Zbigniew Piotrowski — 1997

Mathematica Bohemica

Based on some earlier findings on Banach Category Theorem for some “nice” σ -ideals by J. Kaniewski, D. Rose and myself I introduce the h operator ( h stands for “heavy points”) to refine and generalize kernel constructions of A. H. Stone. Having obtained in this way a generalized Kuratowski’s decomposition theorem I prove some characterizations of the domains of functions having “many” points of h -continuity. Results of this type lead, in the case of the σ -ideal of meager sets, to important statements...

Weak continuity properties of topologized groups

J. CaoR. DrozdowskiZbigniew Piotrowski — 2010

Czechoslovak Mathematical Journal

We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if ( G , · , τ ) is a regular right (left) semitopological group with dev ( G ) < Nov ( G ) such that all left (right) translations are feebly continuous, then ( G , · , τ ) is a topological group. This extends several results in literature.

On the preservation of Baire and weakly Baire category

Alireza Kamel MirmostafaeeZbigniew Piotrowski — 2016

Mathematica Bohemica

We consider the question of preservation of Baire and weakly Baire category under images and preimages of certain kind of functions. It is known that Baire category is preserved under image of quasi-continuous feebly open surjections. In order to extend this result, we introduce a strictly larger class of quasi-continuous functions, i.e. the class of quasi-interior continuous functions. We show that Baire and weakly Baire categories are preserved under image of feebly open quasi-interior continuous...

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