Closure of Union of Singleton Rationals is Real Number Line
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Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.
Let $B_\alpha$ be the singleton containing the rational number $\alpha$.
Then the closure in the set of real numbers $\R$ of the union of all $B_\alpha$ is $\R$ itself:
- $\ds \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha} = \R$
Proof
By definition:
- $B_\alpha = \set \alpha$
Thus:
- $\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$
The result follows from Closure of Rational Numbers is Real Numbers.
$\blacksquare$
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: $1$