Closed normal subgroup

Definition
A closed normal subgroup of a topological group is a subgroup that is both a defining ingredient::closed subgroup (i.e., it is closed as a subset of the whole group with the underlying topology) and a defining ingredient::normal subgroup.

For a Lie group
When the term closed normal subgroup is used in the context of a Lie group, we mean it with respect to the topology induced by the Lie structure.

For an algebraic group
When the term closed normal subgroup is used in the context of an algebraic group, we mean it with respect to the Zariski topology on the underlying algebraic variety.

Facts
When a group has both an algebraic group and a Lie group structure, the notion of closed normal subgroup with respect to the algebraic group structure may be strictly stronger than that with respect to the Lie group structure:


 * Closed normal subgroup for Lie structure need not be closed normal subgroup of algebraic group
 * Quotient map of Lie group structures for algebraic groups need not be quotient map of algebraic groups