[lisa]

Holes in the *geometry* (like the sphere) are okay.

Holes in the *uv map* (like the black spots in the texture) are a problem.

To snap points, there's a "snap" button in Texture Coordinate Editor. You highlight the points you want to move, then click the appropriate snap button. You can snap points together, or you can snap points to the edge.

In your UV map, move the points around until they fill the square. Also make sure they wrap on all sides. ie The point on the left will seam with the opposing point in the right.