In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are ...

The main goal of this paper is to characterize the maximal class with respect to maximums for the family of upper semicontinuous strong Świątkowski functions.

In this paper we characterize the product of internally quasi- -continuous functions and we construct a bounded internally quasi-continuous strong Świa̧tkowski function which cannot be written as a finite product of ...

In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a ...

In this paper, we characterize the maximal additive and multiplicative classes for lower and upper semicontinuous strong Świątkowsk functions and lower and upper semicontinuous extra strong Świątkowski functions. Moreover, ...

In this paper, we present some results concerning functions which are represented as the maximum of two extra strong Świątkowski functions.