Definition:Meager Space

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Definition

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 $T$.


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 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

except the concept is limited to subsets of the real number line