Kuratowski-Zorn Lemma

The Kuratowski-Zorn Lemma is a mathematical principle used in set theory. It states that if you have a partially ordered set (think of a collection where some items can be compared and some cannot), and every chain (a subset where every pair of items is comparable) has an upper bound (a greatest element), then the entire set has at least one maximal element (an item that isn’t surpassed by any other). This lemma is essential for proving the existence of certain types of objects in various branches of mathematics, particularly when direct construction is difficult or impossible.