Continuous images of sets of reals

TL;DR

This paper explores the conditions under which uncountable sets of reals can be continuously mapped onto non-measure-zero sets and examines the properties of their continuous images in Polish spaces.

Contribution

It demonstrates the consistency of uncountable sets mapping onto non-measure-zero sets and identifies uncountable sets with only meager continuous images, revealing new aspects of set mappings.

Findings

Uncountable sets can be continuously mapped onto non-measure-zero sets under certain conditions.

Existence of uncountable sets whose all continuous images into Polish spaces are meager.

The results depend on set-theoretic assumptions and consistency.

Abstract

We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.