Exterior of Union of Singleton Rationals is Empty

Theorem

Let $B_\alpha$ be the singleton containing the rational number $\alpha$.

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology $\tau_d$.

Then the exterior in $\struct {\R, \tau_d}$ of the union of all $B_\alpha$ is the empty set:

$\ds \paren {\bigcup_{\alpha \mathop \in \Q} B_\alpha}^e = \O$

Proof

By definition:

$B_\alpha = \set \alpha$

Thus:

$\ds \bigcup_{\alpha \mathop \in \Q} B_\alpha = \Q$

By definition, the exterior of $\Q$ is the complement of the closure of $\Q$ in $\R$.

By Closure of Rational Numbers is Real Numbers:

$\Q^- = \R$

By Relative Complement with Self is Empty Set:

$\relcomp \R \R = \O$

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: $1$