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.
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