# Definition:Meager Space

## Contents

## 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

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Countability Properties - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**meagre set**