Definition:Baire Category

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Definition

The Baire category of a topological space $T$ is a way of specifying how dense $T$ is:


First Category (Meager)

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.


$A$ is meager in $T$ if and only if it is a countable union of subsets of $S$ which are nowhere dense in $T$.


Second Category (Non-Meager)

$A$ is non-meager in $T$ if and only if it cannot be constructed as a countable union of subsets of $S$ which are nowhere dense in $T$.

That is, $A$ is non-meager in $T$ if and only if it is not meager in $T$.


Also see

  • Results about Baire category can be found here.


Source of Name

This entry was named for René-Louis Baire.


Historical Note

The concept of categorizing topological spaces into meager and non-meager was introduced by René-Louis Baire, during his work to define what is now known as a Baire space.


Sources