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

• Results about Baire spaces can be found here.

Source of Name

This entry was named for René-Louis Baire.