General Topology 19 JAN, 2012

Comparing weak versions of separability

Comparing weak versions of separability

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8].

💡 Research Summary

The paper investigates several weakened forms of separability in topological spaces, focusing on the existence of σ‑discrete dense sets and meager dense sets, as well as their selective (or “choice‑based”) versions. A σ‑discrete dense set is a dense subset that can be written as a countable union of pairwise disjoint open sets, while a meager dense set is dense but belongs to the first category (i.e., it is a countable union of nowhere‑dense sets). The authors introduce selective σ‑discrete separability and selective meager separability: given any sequence of non‑empty open sets, one can pick a point from each so that the resulting selection forms a σ‑discrete dense set (respectively a meager dense set). These selective properties are strictly stronger than their non‑selective counterparts.

The first part of the paper systematically organizes known separability notions and establishes the basic implication diagram: selective σ‑discrete separability ⇒ existence of a σ‑discrete dense set, and selective meager separability ⇒ existence of a meager dense set. The reverse implications fail in general, and the authors set out to construct concrete counterexamples.

A series of classical and specially engineered spaces are examined:

  • Ψ‑spaces (Mrówka spaces) – built from almost disjoint families on ω. They admit σ‑discrete dense sets but do not satisfy selective σ‑discrete separability, showing that the selective requirement can fail even when the non‑selective version holds.
  • Pixley‑Roy spaces PR(X) – the hyperspace of finite non‑empty subsets of X with the Vietoris topology. When X is countable, PR(X) has a σ‑discrete dense set; when X is uncountable, selective σ‑discrete separability breaks down.
  • Sorgenfrey line – the lower‑limit topology on ℝ. It contains meager dense subsets, yet selective meager separability does not hold.
  • Michael line – ℝ with a topology that makes a fixed uncountable set discrete. It is a Baire space with a meager dense set, but again the selective version fails.
  • Alexandroff double‑point spaces, various graph topologies, and modified line constructions are also used to illustrate subtle distinctions.

The authors then explore the influence of set‑theoretic axioms. Under the Continuum Hypothesis (CH) they prove that every space with a σ‑discrete dense set automatically satisfies selective σ‑discrete separability. Conversely, assuming Martin’s Axiom together with ¬CH, they construct a Ψ‑space that has a σ‑discrete dense set but fails the selective version. This demonstrates that the implication “σ‑discrete dense ⇒ selective σ‑discrete” is independent of ZFC.

Answering questions posed by Tkachuk, the paper shows that the statement “if a space has a σ‑discrete dense set then it is selectively σ‑discrete” cannot be settled in ZFC alone; it holds under CH and fails under MA+¬CH. For Hutchinson’s query about meager dense sets, the Michael line and a modified Sorgenfrey line provide counterexamples, proving that the existence of a meager dense set does not guarantee selective meager separability.

The authors also address a problem from reference