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Hello someone know examples of topological spaces of first category in themselves (meager in itself), that are not countable? Note that the Cantor set is nowhere dense in $\mathbb{R}$ but it is not meager in itself, because it is Baire.

Thanks

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The right order topology of $\Bbb R$ satisfies the requirement. The basis elements of this topology are the right rays $B_y=\{x\in \Bbb R\mid x>y\}$. Uncountability is obvious. In this topology, $\mathbb R=\bigcup_{n\in \Bbb N}\{x\in \Bbb R\mid x<n\}$ (left rays). Notice the set $ \{x\in\Bbb R\mid x<n\}$ is nowhere dense by directly appealing to the definition.

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