Continuous images of closed sets in generalized Baire spaces

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.

30 pages