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Main Problem

Given a Lie group.

The connected component of the identity is a Lie subgroup:

  1. It is a subgroup.

  2. 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?

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2  
I don't understand 2. Are you sure you want $\subsetneq$, not $\subseteq$ (in fact you have equality)? The first two properties immediately follow from connectedness and continuity of the maps involved. The fourth follows directly from 1.,2.,3. It remains to ponder 3. – t.b. Aug 18 '12 at 20:59
    
yes, In the question paper it was proper inclusion – Un Chien Andalou Aug 18 '12 at 21:03
2  
Then it is wrong. Was it $\subset$? Note that many people allow equality when writing $\subset$. Consider some examples, like $O(2)$ and $O(3)$, or whatever Lie group you understand well. – t.b. Aug 18 '12 at 21:04
3  
Or simply consider any connected Lie group to see that 2 is wrong. – Alex Becker Aug 18 '12 at 21:05
    
@t.b. yes it was like that. – Un Chien Andalou Aug 18 '12 at 21:06

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