Journal Article

A CONTINUOUS HOMOMORPHISM OF A THIN SET ONTO A FAT SET

MORRIS, SIDNEY A., CHALEBGWA, TABOKA PRINCE

2022

Overview

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\\mathcal {L}^{2}$ , where $\\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\\mathbb {R}$ itself. Finally, it is shown that $\\mathcal {L}$ and $\\mathcal {L}^{2}$ are both homeomorphic to $\\mathbb {P}$ , the space of all irrational numbers.

• Publisher: Cambridge University Press
• Subject: Algebra / Euclidean geometry / Euclidean space / Homomorphisms / Real numbers