[Paper Review] On $Δ$-spaces

Arkady Leiderman, Paul J. Szeptycki|arXiv (Cornell University)|Jul 29, 2023

Advanced Topology and Set TheoryMathematics3 citations

TL;DR

This paper investigates the class of $Δ$-spaces—topological spaces $X$ for which the function space $C_p(X)$ is distinguished—by constructing new examples and counterexamples. It proves that no Souslin tree is a $Δ$-space, establishes consistency results for ladder system spaces under CH, and shows that adding a Cohen real yields a non-$Δ$ ladder system space, resolving several open problems in set-theoretic topology.

ABSTRACT

$Δ$-spaces have been defined by a natural generalization of a classical notion of $Δ$-sets of reals to Tychonoff topological spaces; moreover, the class $Δ$ of all $Δ$-spaces consists precisely of those $X$ for which the locally convex space $C_p(X)$ is distinguished. The aim of this article is to better understand the boundaries of the class $Δ$, by presenting new examples and counter-examples. 1) We examine when trees considered as topological spaces equipped with the interval topology belong to $Δ$. In particular, we prove that no Souslin tree is a $Δ$-space. Other main results are connected with the study of 2) $Ψ$-spaces built on maximal almost disjoint families of countable sets; and 3) Ladder system spaces. It is consistent with CH that all ladder system spaces on $ω_1$ are in $Δ$. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on $ω_1$ which is not in $Δ$. We resolve several open problems posed in the literature.

Motivation & Objective

• To better understand the boundaries of the class of $Δ$-spaces by constructing new examples and counterexamples.
• To investigate whether certain topological spaces—specifically trees, $ψ$-spaces on maximal almost disjoint families, and ladder system spaces—belong to the class $Δ$.
• To resolve open problems from [12], [25], [31], and [32] concerning the structure and properties of $Δ$-spaces.
• To clarify the relationship between $Δ$-spaces, $Q$-sets, and $σ$-discrete spaces, particularly in the context of ZFC and forcing extensions.

Proposed method

• Using the equivalence that $X$ is a $Δ$-space if and only if every countable disjoint family of subsets admits a point-finite open expansion.
• Applying Fodor's lemma to analyze the structure of ordinal spaces and their $G_δ$-topologies, particularly in scattered compact spaces.
• Employing forcing techniques, including adding a single Cohen real, to construct models where specific ladder system spaces fail to be $Δ$-spaces.
• Analyzing point-finite neighborhood assignments and their implications for $G_δ$-sets, especially in scattered Eberlein compact spaces.
• Using PCF theory and CH to construct consistent examples of ladder system spaces in $Δ$, and contrasting them with models where such spaces are not in $Δ$.
• Leveraging known results on $Q$-sets and $Δ$-sets in the reals to inform the topological structure of general $Δ$-spaces.

Experimental results

Research questions

• RQ1 Are all ladder system spaces on $\omega_1$ in the class $\Delta$ under the Continuum Hypothesis?
• RQ2 Can a Souslin tree ever be a $\Delta$-space?
• RQ3 Is there a $\Delta$-space that is not $\sigma$-discrete and has all closed sets $G_\delta$ but is not a $Q$-set space?
• RQ4 Does the existence of a $\Delta$-set imply $2^{\aleph_0} = 2^{\aleph_1}$?
• RQ5 What characterizes scattered compact spaces $W$ such that $W_\delta$ is a $\Delta$-space?

Key findings

• No Souslin tree is a $\Delta$-space, as shown via Fodor's lemma and the structure of limit ordinals with uncountable cofinality.
• It is consistent with CH that all ladder system spaces on $\omega_1$ are $\Delta$-spaces, but this fails in the forcing extension obtained by adding one Cohen real.
• In the Cohen real extension, there exists a ladder system space on $\omega_1$ that is not a $\Delta$-space, demonstrating the sensitivity of the class to set-theoretic assumptions.
• The Alexandroff duplicate of a $Q$-set space is a $\Delta$-space that is not $\sigma$-discrete, yet contains a closed non-$G_\delta$ subset.
• Every subset $F$ of a scattered Eberlein compact space consisting of $G_\delta$ points is itself a $G_\delta$-set, implying $F$ is a $Q$-space.
• A topological space admitting a point-finite neighborhood assignment is a $\Delta$-space, and any $G_\delta$ subset of such a space is again a $G_\delta$-set.