# Zorn lemma

2010 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

maximal principle

If in a non-empty partially ordered set $X$ every totally ordered subset (cf. Totally ordered set) has an upper bound, then $X$ contains a maximal element. An element $x_0$ is called an upper bound of a subset $A\subset X$ if $x\leq x_0$ for all $x\in A$. If an upper bound for $A$ exists, then the set $A$ is said to be bounded above. An element $x_0\in X$ is called maximal in $X$ if there is no element $x\in X$, $x\not=x_0$, such that $x_0\leq x$.

The lemma was stated and proved by M. Zorn in [Zo]. It is equivalent to the axiom of choice.

#### References

 [Ke] J.L. Kelley, "General topology", Springer (1975) MR0370454 [Zo] M. Zorn, "A remark on a method in transfinite algebra" Bull. Amer. Math. Soc., 41 (1935) pp. 667–670 MR1563165