Summarize:

You are likely to run into your ex best friend again sometime. When this happens, you need to be calm and collected. If you can avoid talking altogether, that's great. If not, say hello, and be polite. If you see your old friend at a party and they ask how you have been, respond with something like "I've been doing well. I hope you are well, too." If you run into your former friend at a school event, just acknowledge them and keep going on your way. Being polite does not mean that you have to entertain your ex best friend for a long period of time. Answer any questions briefly, if you are comfortable answering them at all. Avoid asking questions; they only invite more conversation. Be polite if they try to start a casual conversation. For example, if they ask about your mom, say something like "My mom is doing well, thanks." The key here is not to add anything that might continue the conversation, such as "I hope your parents are well," or "How is your mom?" If you have no interest in rekindling your friendship, do not be too accommodating. Be polite in the moment, and leave that moment with closure. If you make it appear as though you miss your ex best friend that opens to the door to starting the friendship and/or fight back up. You don't have to say "It's great to see you," or "See you again soon." This will give them false hope for reigniting the friendship. End the conversation with something like "I'm glad you're doing well, but if you'll excuse me, I need to go talk to my date (or any other person you can think of). Goodbye."
Try to be polite. Keep it short. Leave no room for misinterpretation.