# On the Bourbaki's fixed point theorem and the axiom of choice

## TL;DR

This paper generalizes Moroianu's fixed point theorem, unifies proofs of Bourbaki's fixed point theorem and related results, and demonstrates the equivalence of the axiom of choice with several key lemmas within ZF set theory.

## Contribution

It provides a unified, elegant proof connecting fixed point theorems and the axiom of choice, extending Moroianu's result.

## Key findings

- Unified proof of Bourbaki's fixed point theorem
- Equivalence of the axiom of choice with Kneser's Lemma, Zorn's Lemma, and Zermelo's Lemma
- Generalization of Moroianu's fixed point theorem

## Abstract

In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, the axiom of choice is equivalent to each of the following statements: H. Kneser's Lemma, Zorn's Lemma, Zermelo's Lemma