Discrete Subgroup of Real Numbers is Closed

Theorem

Let $G$ be a subgroup of the additive group of real numbers.

Let $G$ be discrete.

Then $G$ is closed.

Proof

By Subgroup of Real Numbers is Discrete or Dense, there exists $a \in \R$ such that $G = a \Z$.

If $a = 0$, then $G$ is closed.

Let $a > 0$.

Then:

$\ds \R \setminus G = \bigcup_{z \mathop \in \Z} \openint {a z} {a z + a}$

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By Union of Open Sets of Metric Space is Open, $\R\setminus G$ is open.

Thus $G$ is closed.

$\blacksquare$

Also see

• Discrete Subgroup of Hausdorff Group is Closed