Definition:Meager Space

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Definition

Let $T = \left({S, \tau}\right)$ 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 $T$.


Non-Meager

$A$ is non-meager in $T$ if and only if it can not 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 known as

A set which is meager in $T$ is also called of the first category in $T$.


Also see

  • Results about meager (first category) spaces can be found here.


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.


Linguistic Note

The word meager (British English: meagre) is a somewhat old-fashioned word meaning deficient, lacking, scrawny etc.

It originates from the French maigre, meaning thin in the sense of unhealthily skinny.


Sources