  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.
