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 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