# Choice free Fixed Point Property in separable Banach spaces

## TL;DR

This paper demonstrates that fixed point theorems for non-expansive mappings in separable Banach spaces can be proved without the full Axiom of Choice by using descriptive set theory techniques.

## Contribution

It introduces a choice-free approach to fixed point theorems in separable Banach spaces, avoiding reliance on Zorn's Lemma.

## Key findings

- Fixed point theorems hold without the full Axiom of Choice in separable Banach spaces.
- Uses classical and effective descriptive set theory to establish results.
- Provides a new method that simplifies the proof process for fixed point theorems.

## Abstract

We show that the standard approach of minimal invariant sets, which applies Zorn's Lemma and is used to prove fixed point theorems for non-expansive mappings in Banach spaces can be applied without any reference to the full Axiom of Choice when the given Banach space is separable. Our method applies results from classical and effective descriptive set theory.