Baire space
A topological space in which countable intersections of dense open sets are dense
A Baire space is a topological space such that for every sequence of open sets that are dense, the intersection is dense in .
Remarks
This is the setting for Baire category arguments: in a Baire space, “large” sets are often expressed as residual sets, and “small” sets as meager sets (countable unions of nowhere dense sets).
Examples
- Any complete metric space is a Baire space (see complete metric spaces are Baire).
- Any Banach space (with its metric from the norm) is a Baire space.