Baire space

A Baire space is a topological space $X$ such that for every sequence $(U_n)_{n\in\mathbb{N}}$ of open sets that are dense , the intersection $\bigcap_{n\in\mathbb{N}} U_n$ is dense in $X$.

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.