Yeah, if there aren't a lot of mutual friends or social situations where you're going to have to see each other, then it's best to just cold turkey it. And especially if this woman is now engaged to someone else, there's just not much good that will come from trying to interact with her. Most likely, any continued interaction with her at this point is going to look like:
Jerm: "I'm hurt. I want you back."
Her: "I understand. I'm sorry."
Nope, nope, nope. That just prolongs the hurt. Gotta cut her out.
Also, working on improving yourself post breakup is always a great thing to do, but I would caution that one should not approach self-improvement with even the slightest hint of "if I improve myself, I'll get her back some day" in your mind. 9 times out of 10, that just leads to (once again) prolonging the hurt.
Improve yourself for yourself.
Jerm: "I'm hurt. I want you back."
Her: "I understand. I'm sorry."
Nope, nope, nope. That just prolongs the hurt. Gotta cut her out.
Also, working on improving yourself post breakup is always a great thing to do, but I would caution that one should not approach self-improvement with even the slightest hint of "if I improve myself, I'll get her back some day" in your mind. 9 times out of 10, that just leads to (once again) prolonging the hurt.
Improve yourself for yourself.