Group is Hausdorff iff has Closed Discrete Subgroup

Theorem

A topological group is Hausdorff if and only if it has a closed discrete subgroup.

Proof

Necessary Condition

Follows directly from Topological Group is Hausdorff iff Identity is Closed.

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$\Box$

Sufficient Condition

Let $G$ be a Hausdorff topological group.

Let $H \leq G$ be a closed discrete subgroup.

Let $e$ be the identity of $G$.

By Set in Discrete Topology is Clopen, $\left\{{e}\right\}$ is closed in $H$.

By Closed Set in Topological Subspace: Corollary, $\left\{{e}\right\}$ is closed in $G$.

By Topological Group is Hausdorff iff Identity is Closed, $G$ is Hausdorff.

$\blacksquare$

Also see

• Group is Hausdorff iff Discrete Subgroups are Closed