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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): category: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): category: 1.