is the complement of first category is always second category in general space( which is not complete). I think it is true only if the space isx
2026-03-30 01:53:09.1774835589
is the complement of first category is always second category
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Consider $\Bbb Q$ with the usual topology. It has no isolated points, so $\{x\}$ is a closed nowhere dense set for each $x\in\Bbb Q$. $\Bbb Q$ is countable, and $\Bbb Q=\bigcup_{x\in\Bbb Q}\{x\}$, so $\Bbb Q$ is meagre (first category) in $\Bbb Q$. Its complement is empty, which is certainly meagre!