1) Prove the quotient map π : G → G/N is a continuous group-homomorphism.
2) Suppose π: G → G/N is the quotient map and B ⊂ G is an arbitrary set. Prove that π-1(π(B)) = BN. Thus conclude π is an open map.
We will prove G/N is a topological group in two steps, corresponding to the multiplication and inversion maps. The multiplication step relies on the following exercise.
3) Suppose f : A → X and g : B → Y are open maps. Prove the following is an open map:
(f × g): A × B → X × Y (a, b) |→ (f(a), g(b)), when the domain and codomain carry the product topology.
4) Suppose G is a topologial group and N ⊂ G a normal subgroup. Prove that G/N is Hausdorff if and only if N is closed.
5) Let G and F be groups, N ⊂ G be normal, π the canonical quotient map, and φ any function. Consider the mapping diagram.
Prove that φ is a morphism if and only if φ o π is a morphism.
6) Prove the quotient group R/Q has the trivial topology.