Normal core of closed subgroup is closed

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Statement

Suppose G is a topological group and H is a closed subgroup of G. Then, the normal core of H in G (i.e., the largest normal subgroup of G contained in H, also defined as the intersection of all the conjugate subgroups of H) is also a closed subgroup of G.

Proof

Given: A topological group G, a closed subgroup H of G.

To prove: The normal core of H in G is also a closed subgroup of G.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 For any g \in G, the conjugation map x \mapsto gxg^{-1} is a self-homeomorphism of G. Definition of topological group G is a topological group PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
2 For any g \in G, the conjugate subgroup gHg^{-1} is closed in G self-homeomorphisms send closed subsets to closed subsets H is closed in G Step (1) Step-fact-given combination direct
3 The normal core of H in G, which is the intersection \bigcap_{g \in G} gHg^{-1}, is closed in G. arbitrary intersection of closed subsets is closed Step (2) Step-fact combination direct.