Definition:Baire Space (Topology)

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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.


Also see


Source of Name

This entry was named for René-Louis Baire.