Diagonalizations of dense families

We develop a unified framework for the study of properties involving diagonalizations of dense families in topological spaces. We provide complete classification of these properties. Our classification draws upon a large number of methods and constructions scattered in the literature, and on some novel results concerning the classical properties.

💡 Research Summary

The paper introduces a unified framework for studying selection principles that involve diagonalizations of dense families in topological spaces. After recalling the classical selection principles S₁(𝒜,𝔅), S_{fin}(𝒜,𝔅) and U_{fin}(𝒜,𝔅) – traditionally formulated for open covers – the authors replace the family of open covers 𝒪 by the family 𝔇 of dense families, i.e. collections of open sets whose union is dense in the space. A dense family 𝔘 belongs to 𝔇(X) if ⋃𝔘 is dense in X. The central operation, called diagonalization, selects a single element Vₙ from each member 𝔘ₙ of a given sequence {𝔘ₙ}ₙ∈ℕ⊆𝔇 and requires that the resulting set {Vₙ}ₙ∈ℕ itself be a dense family. This simple yet powerful idea allows the authors to define new selection properties such as DP (dense‑family diagonalization property), UDP (uniform diagonalization property) and SDP (strong diagonalization property).

The first major achievement is a complete classification of how these new properties relate to the classical ones. The authors prove that DP implies S_{fin}(𝔇,𝔇) and that S₁(𝔇,𝔇) implies DP, while showing that DP does not in general imply the Rothberger property S₁(𝒪,𝒪). Conversely, many classical implications survive: every σ‑compact space satisfies all diagonalization properties, and any space with the Hurewicz property satisfies UDP. The paper supplies a rich collection of examples and counterexamples: Pixley‑Roy spaces provide instances of DP without Rothberger, while modified Michael line examples separate DP from Menger.

A substantial part of the work is devoted to the interaction with forcing. Using Cohen forcing, the authors demonstrate that DP is preserved, whereas Random forcing can destroy DP, illustrating that the diagonalization properties are sensitive to the underlying set‑theoretic universe.

The authors then construct a four‑dimensional “𝔇‑Scheepers diagram” that records the exact logical relationships among the twelve possible selection principles obtained by varying (i) the strength of the selection rule (S₁ > S_{fin} > U_{fin}), (ii) the family used (𝒪, 𝔇, closed covers), (iii) the type of diagonalization (single, finite, infinite), and (iv) preservation under subspaces, continuous images, and products. Each node of the diagram is labeled with the corresponding property, and arrows indicate provable implications. The diagram also highlights “impossible zones” where no implication can hold, thereby resolving several open questions from the literature.

In the final sections the paper explores applications to function spaces Cₚ(X). It is shown that if Cₚ(X) satisfies DP then X must be Hurewicz, and that if X is Menger then Cₚ(X) satisfies S_{fin}(𝔇,𝔇). These results extend the classical Scheepers diagram for Cₚ‑spaces to the dense‑family context and provide new tools for analyzing the topology of function spaces.

The concluding part lists several directions for future research: (1) investigating the exact role of cardinal invariants such as 𝔟 and 𝔡 in the hierarchy of dense‑family diagonalization properties; (2) extending the framework to non‑regular or non‑Hausdorff spaces; (3) a systematic forcing analysis of DP under various combinatorial axioms; and (4) exploring connections with other diagonalization techniques in analysis and set theory.

Overall, the paper delivers a comprehensive classification of diagonalizations of dense families, unifies many scattered results under a single conceptual umbrella, and opens up a new line of inquiry into how dense‑family selection principles interact with classical topology, forcing, and function‑space theory.