I am sure someone will soon tell you to use the new Paint something to fill the areas in question, applicable in CS4.
Meanwhile, if the paths (do) meet at the centre and extend (at least) as far as the outer circle, you may try a solution that also works in older versions, namely to select everything, and Pathfinder>Divide, then Ungroup the paths.
That will give you closed paths to colour; you may select each path by the outer segment.
What you have shown is the normal fill behaviour of open paths.
The kind of "filling" you describe (creating filled paths in the shape of areas that are merely visually enclosed by several separate paths) is generically known as "flood filling."
Illustrator's treatment of a flood fill tool is its LivePaint feature, introduced in CS3. For instructions on how to use it, look it up in your Online Help.
Instructions on how to use it?
Nick select all the art in this art then click on the Live Paint tool in the Tool Panel and that will turn it into a Live Paint Group.
Now select a color you want to fill a shape and click with the Live Paint Bucket inside that shape it will be highlighted for you.
You also have the option to paint strokes as well and when you place the cursor a path it will change to a Live Paint Brush.
Also you have control over the gap size of paths that are not completely closed.
Object>Live Paint>Gap Options.
Be careful when working with strokes this way as the ends sometimes create an unexpected join.
After coloring the Live Paint Group you can expand it and it will turn it into an ordinary group but will retain all the colors you have given it.