It should not be confused with the space of irrational numbers (sometimes called ‘Baire space’ and coincidentally an example of a Baire space in our sense) or a Baire set? (a subset somewhat analogous to a measurable set but defined by a topological property).
Any open subspace of a Baire space is also a Baire space.
As mentioned above, the space of irrational numbers, or equivalently of infinite sequences of natural numbers, is also known as ‘Baire space’. It is a Baire space in the present sense (since it admits a complete metric), but not much should be made of the fact it has the same name.