Resolvability in products of spaces of small cardinality

TL;DR

The paper advances resolvability theory by showing that the product of two regular isodyne spaces of cardinality $ω_1$ is $ω$-resolvable and, more generally, that the product of any $n+2$ Hausdorff isodyne spaces of cardinality $ω_n$ is $ω$-resolvable. It develops a unifying framework using κ-remote, κ-rare, and κ-disentangled notions, together with carving techniques and a function-based criterion (PResFromFunc) to derive ω-resolvability from well-founded relational constructions. The results hinge on reducing to isodyne factors and constructing dense, pairwise disjoint dense families via intricate combinatorial schemes, including linear fibers and well-founded maps. The work clarifies the limits of these methods at larger cardinals (e.g., ω2) and highlights open questions about resolvability in products of regular vs. non-regular isodyne spaces and in higher-cardinality regimes.

Abstract

We prove that: I. The product of any two regular isodyne spaces of cardinality $ω_1$ is $ω$-resolvable; II. The product of any $n + 2$ Hausdorff isodyne spaces of cardinality $ω_n$ is $ω$-resolvable.

Theorems & Definitions (28)

• Proposition 2.1: J.G. Ceder, Ceder1964
• Corollary 2.2
• Theorem 2.3: Illanes, Bhaskara Rao, Illanes1996BhaskaraRao2019
• Lemma 2.4
• proof
• Definition 2.5
• Proposition 2.6
• proof
• Proposition 2.7
• proof