Zbigniew Piotrowski’s research while affiliated with Youngstown State University and other places

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. Keywordsdevelopability number-feebly continuous-nearly continuous-Novak number-paratopological group-semitopological group-topological group MSC 201054H11-22A05-54C08-54E52

We study continuity points of functions with values in generalized metric spaces. We define the generalized oscillation, which is a useful tool in our study. Let X be a topological space and Y be a weakly developable space. Let ƒ : X → Y be a function. Then the set C (ƒ) of continuity points of ƒ is a Gδ -set in X . Some results concerning continuity points of separately continuous functions as well as functions with closed graphs are also given.

The notions of thin and very thin dense subsets of a product space were introduced by the third author, and in this article we also introduce the notion of a slim dense set in a product. We obtain a number of results concerning the existence and non-existence of these types of small dense sets, and we study the relations among them.

We say that a topological space X is Volterra if for each pair f, g: X→ℝ for which the sets of points at which f, respectively g, are continuous are dense, there is a common point of continuity; and X is strongly Volterra if in the same circumstances the set of common points of continuity is dense in X. For both of these concepts equivalent conditions are given and the situation involving more than two functions is explored.

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.

Our main result is a generalization of the Baire category theorem: if X 1 ,⋯,X k is a finite collection of topological spaces so that X 1 is Baire, and for k>1 each X i except possibly X k has a countable π-base and each X i except possibly X 1 is quasi-regular and strongly countably complete, if 〈C n 〉 is a sequence of separately semi-closed subsets of the product ∏ i=1 k X i and O⊂∏ i=1 k X i is a non-empty open set such that O⊂⋃ n=1 ∞ C n , then there is an integer m such that O∩C ∘ m ≠∅.