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.

$\blacksquare$


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