Real Number is Closed in Real Number Line

Theorem
Let $\left({\R, \tau}\right)$ be the real number line under the usual (Euclidean) topology.

Let $\alpha \in \R$ be a real number.

Then $\left\{{\alpha}\right\}$ is closed in $\left({\R, \tau}\right)$.

Proof
From Open Sets in Real Number Line, the set:
 * $S := \left({-\infty \,.\,.\, \alpha}\right) \cup \left({\alpha \,.\,.\, +\infty}\right)$

is open in $\R$.

Thus by definition of closed, its complement relative to $\R$:
 * $\R \setminus S = \left\{{\alpha}\right\}$

is closed in $\R$.