I've been unable to find an answer to the following question in the literature
on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$
where $\kappa$ is inaccessible. The basic open sets are the $U\_f$ where
$f\in\kappa^{<\kappa}$. A perfect set is a nonempty closed set with no
isolated points. Does a perfect set have cardinality $2^{\kappa}$?
Does it have to have cardinality $>\kappa$?


A variation of an observation in
"Generalized Descriptive Set Theory and Classification Theory"
(arxiv.org/abs/1207.4311)
states that if $T$ is a slim Kurepa tree (as defined in Devlin's
"Constructibility") then there is no continuous (in fact Borel)
injection of $2^{\kappa}$ into $[T]$, but this doesn't answer my question.


Of course, if the existence of a slim Kurepa tree was consistent with
$2^{\kappa}>\kappa^+$ then it would be consistent that my first question
had a "no" answer; I have not been able to find any references on this
question, but I wouldn't be surprised if it were so.



cardinality of perfect sets in generalized Baire space
It is consistent to have a perfect set of size
$\kappa$, or of intermediate size between $\kappa$ and $2^\kappa$. 


To see this, suppose that $\kappa$ is an inaccessible cardinal in $V$, and let
$T=2^{<\kappa}$ be the tree of ${<}\kappa$ binary sequences, and
let $X=[T]={}^\kappa 2$ be the set of branches through this tree in
$V$. So $X$ is a perfect set in $V$ and has size $2^\kappa$ there.


Next, force over $V$ to add a Cohen real $c\subset\omega$ and then
to collapse $(2^\kappa)^V$ to $\kappa$. Consider the resulting
forcing extension $V[c][G]$.


The combined forcing admits a closure point at $\omega\_1$, and it
follows from lemma 13 of my article [Extensions with the approximation and cover properties have no new large cardinals](http://jdh.hamkins.org/approximation-and-cover-properties/)
that the extension $V[c][G]$ does not add any new branches through
a tree of height $\kappa$ in the ground model $V$. In particular, $X$ is still the
set of all branches through $T$ in $V[c][G]$. Note that $X$ still
has no isolated points. Further, since there are no new branches
through $T$ in $V[c][G]$, it follows that $X$ is still closed (since any new element of the closure of $X$ would have to be a branch through $T$).


So $X$ is perfect in $V[c][G]$, but since $(2^\kappa)^V$ was
collapsed to $\kappa$, it follows that $X$ has size $\kappa$
there.


One can modify the argument to make a perfect set of size between
$\kappa$ and $2^\kappa$. Start in $V$ with $2^\kappa=\kappa^+$,
but now force to add a Cohen real and then let $G$ pump up
$2^\kappa$ very high. The set $X=({}^\kappa 2)^V$ will still be
perfect in $V[c][G]$, but now it will have size $\kappa^+$, even
though $2^\kappa$ is large in $V[c][G]$. One can make it have size
$\kappa^{++}$ or whatever you like, in the same way.


