Definition:Meager Space
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
- 1973: Thomas J. Jech: The Axiom of Choice ... (previous) ... (next): $1.$ Introduction: $1.4$ Problems
- except the concept is limited to subsets of the real number line
- 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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): category: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): category: 1.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): meagre set