Surjections and weak forms of The axiom of choice

Abstract

If there is an injection from a nonempty set A into a set B, then there is always a surjection from B onto A. Without the Axiom of Choice (AC), the converse of this statement is not necessarily true. There are many other consequences of AC concerning surjections. We study these consequences and introduce some new weak forms as well as equivalent forms of AC and show some relationships among them.