Union of Closures of Singleton Rationals is Rational Space
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Theorem
Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$.
Let $B_\alpha$ denote the singleton containing the rational number $\alpha$.
Then the union of the closures in the set of real numbers $\R$ of all $B_\alpha$ is $\Q$:
- $\ds \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} = \Q$
Proof
Let $\alpha \in \Q$.
By Real Number is Closed in Real Number Line, $B_\alpha = \set \alpha$ is closed in $\R$.
From Closed Set Equals its Closure, it follows that:
- $B_\alpha = \map \cl {B_\alpha}$
Hence the result.
$\blacksquare$
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: $1$