Continuous Images of Closed Sets in Generalized Baire Spaces

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