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I'm getting confused by the following definition and examples:

A Baire space is a topological space in which any meagre set has dense complement, where a set is meagre or of first Baire category if it is a countable union of subsets which are nowhere dense in the space. A set is nowhere dense if its closure has empty interior.

The real numbers, with the usual topology, is a Baire space. The rational numbers, with the usual topology, is not a Baire space.

Could anyone show me how the two examples could be verified using the definitions? I'm having troubles seeing the properties of sets in the two spaces. Thanks.

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1 Answer

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The set of all rationals numbers can be written as the union of singleton sets.These sets are nowhere dense sets, so $\mathbb Q$ is itself meager and its complement is empty. It follows that $\mathbb Q$ is is not a Baire space.

The fact that $\mathbb R$ is a Baire space follows from Baire Category Theorem.

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