= Solution
\b[The continuum hypothesis holds.] Let $A$ be the assumed <Lusin set> of <cardinality> $\mathfrak c=2^{\aleph_0}$. If $\mathfrak c>\aleph_1$, choose $B\subseteq A$ with $|B|=\aleph_1$. The size assumption makes $B$ a <meagre set>, but then $A\cap B=B$ is uncountable, contradicting the <Lusin set> property. Thus
$$
\boxed{2^{\aleph_0}=\aleph_1}.
$$
One can also use (d) directly: <well-order> $A$ in type $\mathfrak c$. Each proper initial segment has size below $\mathfrak c$, so is meagre and consequently countable, since it is a subset of the <Lusin set>. These initial segments form a <linear increasing union of countable sets> equal to $A$. The bound in (d) gives $\mathfrak c\leq\aleph_1$.
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