Real Number Line is Separable
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Theorem
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is separable.
Proof 1
The rational numbers $\Q$ form a metric space.
We have that the Rationals are Everywhere Dense in Topological Space of Reals.
We also have that the Rational Numbers are Countably Infinite.
The result follows from the definition of separable space.
$\blacksquare$
Proof 2
Follows from:
$\blacksquare$
Sources
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): separable space