Normal core of closed subgroup is closed
Suppose is a topological group and is a closed subgroup of . Then, the normal core of in (i.e., the largest normal subgroup of contained in , also defined as the intersection of all the conjugate subgroups of ) is also a closed subgroup of .
Given: A topological group , a closed subgroup of .
To prove: The normal core of in is also a closed subgroup of .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||For any , the conjugation map is a self-homeomorphism of .||Definition of topological group||is a topological group||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]|
|2||For any , the conjugate subgroup is closed in||self-homeomorphisms send closed subsets to closed subsets||is closed in||Step (1)||Step-fact-given combination direct|
|3||The normal core of in , which is the intersection , is closed in .||arbitrary intersection of closed subsets is closed||Step (2)||Step-fact combination direct.|