The question is: If is a continuous map and there is such that (ball center , radius r) for every , then .
I have an idea of a proof but I'm not sure how to formalize it using liftings, etc (I'm not well-practiced in doing calculations with non-trivial fundamental groups). After supposing there is in such that , my idea is to consider the standard loop in , , but with radius and centered at , call it . Each point has image under contained in , and so is contained in the annulus, center a, with inner radius r, outer radius 3r. Put . Since is continuous and is homotopic to the constant loop at , should be homotopic to a constant loop. But joining points of in the balls around the four axes-intercepts , for example, (and perhaps winding around again to be certain), we can see that has to wind around . So since , it isn't homotopic to a constant loop, a contradiction.
It's frustrating that I have the idea but I am not not familiar enough with performing calculations to formalize this. If someone could help me get started I'd be most appreciative!