In topology, a **Baire space** is a topological space that satisfies a specific property relating to the completeness of the space in a certain sense.
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A Baire space is a topological space in which every countable intersection of open dense sets is dense, equivalently one in which no nonempty open set is a countable union of nowhere-dense sets. Every compact Hausdorff space and every complete metric space is a Baire space.