Year: 2025
Type: article
Abstract: We prove that in every nonempty perfect Polish space, every dense Gδ subset contains strictly decreasing and strictly increasing chains of dense Gδ subsets of length c, the cardinality of the continuum. As a corollary, this holds in Rn for each n≥1. This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality c, each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set T of real Mahler T-numbers is a dense Borel set and contains a strictly descending chain of length c of proper dense Borel subsets.
DOI: 10.3390/ijt2040020
Author: Sidney Allen Morris
Institutions: Federation University, La Trobe University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,207
OpenAlex ID: W4416913014