A lot of people try to stay friends after a relationship breaks up. While many try, most fail. I believe post-breakup relations are primarly the result of how the relationship ended. If the couple doesn’t have much contempt for each other, then I think a friendship is a real possibility. Both can think of each other lovingly; thinking of the other person doesn’t drudge up bad feelings and memories.
Also, a lot depends on who did the dumping. Obviously, a mutual breakup is best but that doesn’t happen very often. If the person who did the dumping had the power during most of the relationship, I’d say the relationship post-breakup is doomed. The person who got dumped likely has a lot of contempt for the dumper/wants to get back into the relationship.