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September 16, 2025Open Access

Chains of Dense G\ (_ \) Sets in Perfect Polish Spaces

Puntos clave

In perfect Polish spaces, every dense G_delta subset contains dense chains of length equal to the continuum.
The result shows a chain of dense subsets exists in spaces like ℝⁿ with cardinality of the continuum.
Counterexamples reveal the necessity of both perfection and Polishness for the findings to hold.
Dense Borel sets, like the real Mahler T-numbers, exhibit strictly descending chains of dense subsets.

Resumen

We prove that in every nonempty perfect Polish space, every dense \ (G_ \) subset contains strictly decreasing and strictly increasing chains of dense \ (G_ \) 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.

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Cite This Study

Sidney A. Morris (Fri,) studied this question.

synapsesocial.com/papers/68d4506b31b076d99fa57875 https://doi.org/https://doi.org/10.20944/preprints202509.1088.v1