Baire space

A topological space (in particular, a metric space ) $X$ is a Baire space if for every sequence of open dense sets $U_1,U_2,\dots\subseteq X$, the intersection $\bigcap_{n=1}^\infty U_n$ is dense in $X$.

Equivalently, $X$ is a Baire space if and only if:

• No nonempty open set in $X$ is meager (i.e., every nonempty open set is of second category in itself).

In analysis, Baire spaces matter because they make “generic” (residual ) properties meaningful: residual sets are automatically dense.

Examples:

• Every complete metric space is a Baire space (Baire Category Theorem).
• $\mathbb{R}^k$ with the Euclidean metric is a Baire space.

Non-example:

• A metric space that is a countable union of nowhere dense sets is not a Baire space (by definition).