Generalized Luzin sets

TL;DR

This paper explores generalized (I,J)-Luzin sets, extending classical Luzin and Sierpinski sets, providing conditions for their existence, constructing many non-equivalent examples, and identifying forcings that preserve their properties.

Contribution

It introduces the concept of generalized (I,J)-Luzin sets, establishes conditions for their existence, and analyzes their preservation under certain forcings.

Findings

Set-theoretical conditions for existence of (I,J)-Luzin sets

Construction of large families of non-equivalent (I,J)-Luzin sets

Identification of forcings that preserve (I,J)-Luzin property

Abstract

In this paper we invastigate the notion of generalized (I,J) - Luzin set. This notion generalize the standard notion of Luzin set and Sierpinski set. We find set theoretical conditions which imply the existence of generalized (I,J) - Luzin set. We show how to construct large family of pairwise non-equivalent (I,J) - Luzin sets. We find a class of forcings which preserves the property of being (I,J) - Luzin set.