Real Number is Closed in Real Number Space

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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 $\R$.

$\blacksquare$