Since no one is posting I'll post the first thing I though of (Note, I don't know the first thing about topology, well maybe a little metric spaces, so this is not a proof or anything... Just a though).
Consider the discrete dynamical system.
Some may recognize this as the formula to generate the Mandelbrot set when k=2. But if you increase k you get what's known as multibrot sets. As k diverges to infinity the multibrot sets converge to the unit disk.
So my thought was that since the Mandelbrot set is connected (although I'm pretty sure this proof involves the unit disk being connected lol...) perhaps that implies the multibrot sets are connected and taking k to infinity will show that the unit disk is therefore connected.
Just my thought...