Definition:Meager Space/Non-Meager

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A \subseteq S$.

Then $A$ is non-meager in $T$ iff it can not be constructed as the union of a countable set of subsets of $S$ which are nowhere dense in $T$.

That is, $A$ is non-meager in $T$ iff it is not meager in $T$.

Also known as
A subset which is non-meager in $T$ is also referred to as of the second category in $T$.

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.

Historical Note
The concept of categorizing topological spaces in this way was introduced by Baire, during his work to define what is now known as a Baire space.