Discrete subgroup implies closed
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
- Every open subset of a topological group containing the identity has a symmetric open squareroot: If is an open subset of a topological group containing the identity element of the topological group, there exists an open subset such that is symmetric: it contains the identity element and is closed under the inverse map, and further, such that
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.