Definition:Baire Space (Topology)

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Then $T$ is a Baire space if the union of any countable set of closed sets of $T$ whose interiors are empty also has an empty interior.

Equivalent Definitions
This definition is equivalent to each of the following conditions:


 * $(1): \quad$ The intersection of any countable set of open sets of $T$ which are everywhere dense is everywhere dense.


 * $(2): \quad$ The interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty.


 * $(3): \quad$ Whenever the union of any countable set of closed sets of $T$ has an interior point, then one of those closed sets must have an interior point.

Also see

 * Equivalence of Definitions of Baire Space