My girlfriend is still talking to her ex behind my back after she promised me she never would again..

Anonymous
I have been dating my girlfriend for a little over six months. She was dumped by her ex about 3 months before we got together. When we first got together, I knew she still had feelings for him, but as time went on, she seemed much happier and I could tell she was getting over him.

Recently she left her Facebook on at my house and I saw she had been talking to her ex and saying things like she missed him, and he said how his new girlfriend didn't make him happy like she did, and she responded saying that she thinks I'm great but it isn't the same...

She tells me she loves me everyday, we have a great sex life, and she is extremely happy around me and anxious to spend lots of time with me.

But I couldn't handle that and told her it was over, and she begged me to take her back and promised she wouldn't do it ever again. She blocked him on Facebook, and now about a week later. she was at my house checking her email and I she had been chatting with him over email.

I feel like I should just break up with her for good, but I'm not sure if she is just confused and really loves me, or I'm just a space filler.

Thoughts?
Updates
+1 y
I agree with all of you, but I'm just totally baffled how she tells me she wants to have my kids and marry me and does this. A little note, the second time she emailed him, it was just one message saying that I made her block him on Facebook and she is sorry, and if he needs anything he should message her, but she isn't actively engaging him anymore
My girlfriend is still talking to her ex behind my back after she promised me she never would again..
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