Separate continuity topology and a generalization of Sierpinski's theorem

TL;DR

This paper explores the properties of the separately continuity topology and uses these insights to generalize Sierpinski's theorem, showing that real separately continuous functions can be determined by their values on dense sets.

Contribution

It introduces a generalization of Sierpinski's theorem using properties of the separately continuity topology.

Findings

Properties of the separately continuity topology are characterized.

A generalized Sierpinski theorem for real separately continuous functions is established.

Functions are determined by their values on arbitrary dense sets.

Abstract

The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an arbitrary dense set is obtained.