Group is Connected iff Subgroup and Quotient are Connected
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Theorem
Let $G$ be a topological group.
Let $H \le G$ be a subgroup.
The following statements are equivalent:
- $(1):\quad$ $G$ is connected
- $(2):\quad$ $H$ is connected and the left quotient space $G / H$ is connected
- $(3):\quad$ $H$ is connected and the right quotient space $G / H$ is connected.
Proof
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