# Definition:Baire Space (Topology)

## Definition

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

### Definition 1

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

### Definition 2

$T$ is a Baire space if and only if the intersection of any countable set of open sets of $T$ which are everywhere dense is everywhere dense.

### Definition 3

$T$ is a Baire space if and only if the interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty.

### Definition 4

$T$ is a Baire space if and only if, 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.

## Source of Name

This entry was named for René-Louis Baire.