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.


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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