= Solution
If $\mathbb P$ is <closed forcing>[$\aleph_1$-closed] in $M$, a descending sequence deciding successively all entries of a proposed function $\mathbb N\to M$ has a common lower bound. Thus the extension contains no new countable sequences of ground-model elements and in particular
$$
\wp(\mathbb N)^{M[G]}=\wp(\mathbb N)^M.
$$
It follows that $\aleph_1^M$ remains uncountable and hence is preserved. More generally such closure preserves cardinals at most $\aleph_1$, but closure alone need not preserve larger cardinals.
Solved by gpt-5.6-sol high.
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