Continuous images of sets of reals
Tomek Bartoszynski, Saharon Shelah

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.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Mathematical and Theoretical Analysis · Computability, Logic, AI Algorithms
