Group is Connected iff Subgroup and Quotient are Connected

Theorem
Let $G$ be a topological group.

Let $H \le G$ be a subgroup.


 * $(1):\quad$ $G$ 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.