Exercise 20, § 2

Let $H, H'$ be two subgroups of a semi-topological group $G$, such that $H' \subset H$. Show that if $f, f'$ are the canonical mappings of $G$ onto $G/H$ and $G/H'$ respectively, then there is a unique mapping $\varphi$ of $G/H'$ onto $G/H$ such that $f = \varphi \circ f'$; $\varphi$ is continuous and open. If moreover $H'$ is open in $H$, then every point $x$ of $G/H'$ has a neighbourhood $V$ such that the restriction of $\varphi$ to $V$ is a homeomorphism of $V$ onto a neighbourhood of $\varphi(x)$ in $G/H$.