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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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If $\{X_i\}_{i \in I}$ is a family of topological spaces, let $X = \coprod_{i \in I} X_i$ be their coproduct (a.k.a. disjoint union). Then $X$ is meager iff each $X_i$ is meager in itself (iff each $X_i$ is meager as a subset of $X$). This is not hard to prove, using the observation that a subset $Y$ of $X$ is nowhere dense iff its intersection to each $X_i$ is nowhere dense.

It follows from this that for every infinite cardinal $\kappa$, the coproduct $\coprod_{i \in \kappa} \mathbb{Q}$ of $\kappa$ copies of the rational numbers is a meager topological space of cardinality $\kappa$. These spaces are all metrizable, as a coproduct of metrizable spaces is metrizable.

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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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