So now that you two have broken up, what is there for the both of you? I'll tell you what, NOTHING! That is absolutely right. After someone breaks up with you, or you break up with them (whichever makes you sleep easier at night), then there should be nothing between the two of you. One of you might suggest to remain friends. Any relationship counselor will tell you that is a terrible idea. Friendships rarely work between exes for a long time after a break up. And if you've committed to a new relationship with someone else while still talking to the ex, you're headed for absolute disaster. AND you're being disloyal to your new partner.
So why do certain people try to remain friends after break-ups? Some do it because although they want to end the relationship, they still deeply care for the person they just ended it with, and so want to be in the know of what is happening. That is a terrible idea because it can only lead to heartache when that person moves on from you and decides to start dating/sleeping with someone else. Some people do it because they actually like hanging out with the person things are ending with, and despite the fact that things are ending, they want to try to keep that friendship going. However, that is just another way of keeping feelings going between the two of you, and also leads only to heartache and ultimately ruins the new relationship you've just started.
When you break up with someone, you need to cut that person out of your life for a while. There is no way for you to get over someone if you are in constant contact with them. NO WAY AT ALL.
Friendships can be renewed after a break up but that is only way down the road, and this is only after you have moved beyond that person. You have to be 100% positive that no feelings remain for that person in order to try to become friends again. Sexual tensions can linger because, well lets be honest, sex can be purely physical and not involve feelings at all. However, if the feelings still remain in any way whatsoever then a friendship is simply out of the question.
Time is required to heal and move on, as well as distance. Avoiding contact with the ex is always the only choice, and if he or she tries to hang on and insists on contact, it's a big red flag and you have to get rude and put your foot down. If you don't, you're just as guilty as they are for making the breakup harder than it needs to be and if you're in a new relationship the only right thing to do is immediately cut off the old flame.
Remember that next time they suggest to remain "friends", or you are the person dumb enough to suggest to remain friends. Because friendships between exes don't work, at least not until a year or so down the road.
If you don't have the willpower to do this, then don't bother trying a new relationship, because you're heading down a road of heartache, lies and betrayal and it's not fair for you, your ex, or your new partner.