Zorn's lemma

Zorn’s lemma is a very useful tool to talk about the existence of a maximal element (this can be used for maximal subgroups, tori, etc. where the lemma applies).

Statement

Let be a non-empty partially ordered set.
If every chain (a subset of that is totally ordered under the induced ordering, i.e. a relationship exists for every pair in the subset) has an upper bound in , then has at least 1 maximal element.

Examples

Consider the real numbers . If we have a set , then it is clear that every chain in the interval has a upper bound that is in . Therefore, there is a maximal element (in this case ).