Meager topological spaces of uncountable size

Question

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

Answer 1

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.

Answer 2

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.