Rational Number Space is Meager
Jump to navigation
Jump to search
Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is meager.
Proof
From Rational Numbers are Countably Infinite, $\Q$ is a countable union of singleton subsets.
From Singleton Set is Nowhere Dense in Rational Space, each of those singleton subsets is nowhere dense in $\struct {\Q, \tau_d}$.
The result follows from definition of meager.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30$. The Rational Numbers: $6$
- 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.