Discrete subgroup implies closed

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This article gives the statement and possibly, proof, of an implication relation between two topological subgroup properties. That is, it states that every subgroup of a topological group satisfying the first subgroup property must also satisfy the second
View a complete list of topological subgroup property implications

Statement

Any discrete subgroup of a T0 topological group (i.e., a subgroup that is discrete in the subspace topology), is a closed subgroup.

Facts used

Proof

Given: A T0 topological group , a discrete subgroup

To prove: is a closed subgroup of

Proof: Let denote the identity element of . Since is discrete, there exists an open set such that . By the fact stated above, there exists a symmetric open subset such that .

Now, suppose is not closed. Then there exists an element such that every open subset containing intersects . This yields that every open subset containing the identity intersects . In particular, intersects . Note that since does not intersect , . Hence, . Thus, is also an open subset containing the identity, and hence it again intersects . Thus, we can find another point .

Now consider . This is an element of . Moreover, since is symmetric, , so . Finally, since , . Thus, we have found a non-identity element in , a contradiction.