Real Number is Closed in Real Number Line

Theorem

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

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

Then $\set \alpha$ is closed in $\struct {\R, \tau}$.

Proof

From Open Sets in Real Number Line, the set:

$S := \openint \gets \alpha \cup \openint \alpha \to$

is open in $\R$.

Thus by definition of closed, its complement relative to $\R$:

$\R \setminus S = \set \alpha$

is closed in $\R$.

$\blacksquare$