Main Problem
Given a Lie group.
The connected component of the identity is a Lie subgroup:
It is a subgroup.
It is open.
How to check this using topological tools?
Extra Problem
The quotient by the above is the group of connected components: $$G_e\lhd G:\quad (gG_e)(hG_e)=(ghG_e)=G_{gh}$$
How to check this using topological tools?