Continuous images of closed sets in generalized Baire spaces

TL;DR

This paper investigates the structure of continuous images of closed sets in generalized Baire spaces, revealing complex relationships and independence results within set theory.

Contribution

It establishes new separation results among classes of continuous images in generalized Baire spaces and demonstrates independence from ZFC axioms.

Findings

Existence of closed sets not continuous images of ${}^$

Existence of injective continuous images not $$-Borel

Independence of certain statements from ZFC axioms

Abstract

Let $\kappa$ be an uncountable cardinal with $\kappa=\kappa^{{<}\kappa}$. Given a cardinal $\mu$, we equip the set ${}^\kappa\mu$ consisting of all functions from $\kappa$ to $\mu$ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$. We prove results that allow us to separate several classes of subsets of ${}^\kappa\kappa$ that consist of continuous images of closed subsets of spaces of the form ${}^\kappa\mu$. Important examples of such results are the following: (i) there is a closed subset of ${}^\kappa\kappa$ that is not a continuous image of ${}^\kappa\kappa$; (ii) there is an injective continuous image of ${}^\kappa\kappa$ that is not $\kappa$-Borel (i.e. that is not contained in the smallest algebra of sets on ${}^\kappa\kappa$ that contains all open subsets and is closed under $\kappa$-unions); (iii) the statement "every continuous image of ${}^\kappa\kappa$ is an injective continuous image of a closed subset of ${}^\kappa\kappa$" is independent of the axioms of $\mathrm{ZFC}$; and (iv) the axioms of $\mathrm{ZFC}$ do not prove that the assumption "$2^\kappa>\kappa^+$'' implies the statement "every closed subset of ${}^\kappa\kappa$ is a continuous image of ${}^\kappa(\kappa^+)$'' or its negation.