I use Illustrator CS4. A closed path means a path that is a completed shape - when you are finished, there should no longer be a beginning and and end. Envision a circle - no beginning or end. This is especially important when you are using a fill color or gradient. To close a path, 1) using your pen tool, select one of the end points. 2) move your mouse to the other end point in your path. As you get close, there will be an icon that appears next to your pen that looks like a little circle. When you see that icon, click on the end point, and that will complete the path. If you just click on the end point, a straight line will connect the beginning and the end of the path. If you click and drag, a curved line is the result, and where you drag determines the shape of that curved line.