R4L,
I think this depends on a few factors. If the girl ended the relationship with you, then there usually isn't much reason to keep in contact. It means, in some way, you messed things up with her to the point where she felt like she needed to end it, and attempting to stay around a girl who doesn't want to be around you anymore is usually just going to be more drama and emotional tugging than anything else. Although, these days, girls never really end their relationships with me, so it's not something I have to worry about.
If it's a girl you broke it off with, then I think it really depends on how you broke it off with her. If you can end things in a way that communicate to her that you're moving on but you're still willing to be there for her as a friend, then you can make it work that way. But it also depends on whether or not you see any value in her as a friend because, if there isn't any, then she might just be keeping her hopes up for some type of future reunion when it may never happen.
Most relationships end in some type of emotional, traumatic break-up that results in it being difficult for post-relationship "friendship" to actually work, so if that's the way that a particular relationship ended with a girl, then I would probably just let her go.
- Franco