Here is a picture of the set $F$ (in red) in my comment above (the black lines represent the sphere is of radius 2). There is a component of $A$ inside the sphere and a component outside the sphere.
Here is a picture of the set $F$ in my comment above (the sphere is of radius 2). There is a component of $A$ inside the sphere and a component outside the sphere.
Here is a picture of the set $F$ (in red) in my comment above (the black lines represent the sphere of radius 2). There is a component of $A$ inside the sphere and a component outside the sphere.
Here is a picture of the set $F$ in my comment above (the sphere is of radius 2). There is a component of $A$ inside the sphere and a component outside the sphere.