Here is what I think:

Say I have open sets X and Y, then for , I can find an open ball centered at them such that the entire ball is in X and Y, respectively.

Well, then, if you pick any point in , that that point is either in X or Y, well, then, you can draw another open ball that is contained in either X or Y.

So if you use radius for the points in X and Y, you can just use radius

Hope this helps.