Let $F$ be a countably union of $F_\sigma$ sets, say $F_i$ for $i\in \mathbb N$. Is it true that if $F$ is not meager then there is an $F_i$ with non-empty interior? For me it's true but I can't find a strong argument to prove it. Thank you
2026-03-30 01:51:39.1774835499
on a countably union of $F_\sigma$ sets
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You can use contradiction to prove it. If all $F_i$ are of empty interiors, then countable union of $F_i$ is meager.