The Chicago White Sox share the town with the Chicago Cubs. That fact has led to a pretty big rivalry. The rivalry is pretty much a fan-based rivalry, however, and it doesn’t impact either team on the field that much. They have the crosstown series and that is it. They are not in the same division or league for that matter, so it doesn’t really matter in the grand scheme of things. They will never impact each other’s chances of winning the World Series unless they actually play in the fall classic.
They have proven that point in recent years as they made a huge trade with each other in 2017.