Should I email him a closure letter?

No. It will make you seem desperate to communicate. He's made his wishes known, you know how he feels so just leave it alone. It's difficult and closure would be nice but it will not come off well. Get closure in your own mind, be mad if you must but he doesn't need to know you're still hurting.

Oh get real, you and everybody else knows a "closure letter" is just an excuse to stay in contact with him and try to say something that will get him to respond. You say you broke it off? Broke what off? There was no boyfriend/girlfriend relationship. It was just in your head. He seems to have always told you in the future, or no he'd cheat. He never ever seems to have said you were his girlfriend. So I'm unclear on what you thought you were breaking up. You need to move along, forget about it, and chase someone else. No letter.

Closure is something only needed if you wronged somebody else (IMO), for example to apologize or explain your actions or seek forgiveness. It sounds like he wronged you and a letter is an attempt to maintain a connection (not in a healthy way). Write this letter to yourself, seal it, burn it, flush it, put it in a box tucked away somewhere for good. You are too good for him and he doesn't need or deserve the letter. Let him go. Walk away with class and dignity. His loss. Best wishes to you.

It doesn't sound as if he needs closure...and anyway how did he lie to you and cheat on you if he told you straight up that he didn't want to hurt you and that he would end up cheating on you..sounds like he told you the truth from the beginning. If you need to write it down then do so, but don't mail it - he's not the one that needs closure - you do.

dont do anything. just leave him alone,, it may be hard at first but if he does not want you, you cant change that.