In a Polish space, a set is comeager if it is nonmeager in every open set?
I read this from a proof of a paper. I'm quite new for the meager set so I didn't grasp it.
2026-03-28 13:21:32.1774704092
In a Polish space, a set is comeager if it is nonmeager in every open set?
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No (assuming the Axiom of Choice). For example, in $\mathbb R$ a Vitali set is nonmeager in every open set, but is not comeager.
EDIT: On the other hand, if $A$ is nonmeager in every open set and has the property of Baire, there is an open set $U$ such that the symmetric difference $A \Delta U$ is meager. Then $U$ must be dense, and $A$ is comeager.