In a Baire space $X$, if an open set $U$ meets a nonmeager set $N$, is the intersection nonmeager?
If they do not meet then it's false; take the upper half line of the reals. It does not meet some of the open sets below $0$.
2026-03-30 06:50:13.1774853413
In a Baire space $X$, if an open set meets a nonmeager set, is the intersection nonmeager?
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No. The nonmeagerness of the nonmeager set may happen somehwere away from where it meets the open set (which might be even just a single point).