  This paper presents relations between several types of closedness of a
law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In
particular, we show that order closedness,
$\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and
$\sigma(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in
$\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order
continuous dual of $\mathcal{X}$. We also provide some application to proper
quasiconvex law-invariant functionals with the Fatou property.
