Given a locally path connected topological group, I'm asked to show that if $G_0$ is the connected component of identity then there is a homeomorphism between $G/G_0$ and $\pi_0(G)$ (which I imagine is the space of path-connected components of $G$ which quotient topology.)
My intuition says that, more than a homeomorphism, the path connected-components are preciesly the cosets of $G_0$. I know for example that every coset of $G_0$ is contained in a component (as multiplication by a fixed element is continuous and must send a a connected set into a connected set.)
I'm using connected and path-connected interchangeably as I think they coincide for locally path connected spaces.
Can anybody guide me towards a solution?
Hint: For any $g\in G$, left multiplication by $g$ is a homeomorphism $G\to G$. Since $G_0$ is the connected component of the identity, what does this tell you about $gG_0$?