Connectedness of sets acting on topological groups...

Question

I come now with a topological group question. Suppose a topological group $G$ acts on a topological space $X$. Suppose $G$ and $X/G$ are connected. Show $X$ is connected. Me and a few friends have been pondering this one for awhile to no avail.

Answer 1

The map $q:X\to X/G,\ x\mapsto G\cdot x,$ is a quotient map because we equip $X/G$ with the quotient topology.

One can prove that if $q:X\to Y$ is a quotient map and $Y$, as well as all the fibers $q^{-1}(y),\ y\in Y$ are connected, then $X$ is connected, too, see my answer here.

In your case the fibers $q^{-1}(G\cdot x)$ are just the orbits $G\cdot x$. Since $m:G\times X\to X,\ (g,x)\mapsto g\cdot x$ is continuous, we see that $G\cdot x=m(G\times\{x\})$ is connected, being the continuous image of a space homeomorphic to $G$.