Group is Connected iff Subgroup and Quotient are Connected

Theorem
Let $G$ be a topological group.

Let $H \le G$ be a subgroup.

Then the following are equivalent:
 * $G$ is connected
 * $H$ is connected and the left quotient space $G / H$ is connected
 * $H$ is connected and the right quotient space $G / H$ is connected.