Definition:Meager

Definition

Let $\left({S, \vartheta}\right)$ be a topological space.

Let $A \subseteq S$.

Then $A$ is meager in $S$ iff it is the union of a countable set of subsets of $S$ which are nowhere dense in $S$.

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

Non-Meager

A set which is not meager is called non-meager in $S$ or of the second category in $S$.

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.

Also see

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

• Results about non-meager (second category) spaces can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Countability Properties