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Continuous images of closed sets in generalized Baire spaces (2302.01006v1)

Published 2 Feb 2023 in math.LO

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.

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