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By being of the second category I mean that it is not the countable union of nowhere dense sets and by Baire space I mean a space such that a countable intersection of open dense sets is dense in X.

It's easy to see that if it is a Baire space then it is of the second category, but I can't see why the converse would hold.

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Let $Y$ be any Baire space, and let $X$ be the disjoint union $Y\sqcup\Bbb Q$, where $\Bbb Q$ has its usual topology. (In other words, $Y$ and $\Bbb Q$ are clopen subsets of $X$.) $X$ cannot be the countable union of nowhere dense subsets, since $Y$ cannot, but $\big\{X\setminus\{q\}:q\in\Bbb Q\big\}$ is a countable family of dense open sets whose intersection, $Y$, is not dense in $X$.

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