= Closed subgroup
A <closed subgroup> is a <subgroup> which is a <closed set> in the ambient <topology>. It is a <topological group> in its <subspace topology>, but need not be an <open subgroup>.
The additive <subgroup> $\mathbb Z\subset\mathbb R$ of <integers> is a <closed subgroup>: every noninteger lies in an <open interval> $(n,n+1)$ avoiding $\mathbb Z$. It is also a <discrete subgroup>, since $(n-1/2,n+1/2)\cap\mathbb Z=\{n\}$. Nevertheless it is not an <open subgroup> of $(\mathbb R,+)$: every <open interval> containing an <integer> contains nonintegers too. Thus having a <discrete topology> internally says nothing about being an <open set> in the ambient <topology>.
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