Examples of two definitions equivalent in ZFC, but inequivalent in ZF set theory.

Question

What are two definitions of an object or class of objects, which are equivalent assuming the axiom of choice, but are inequivalent assuming only ZF set theory? I am not looking for theorems that are inequivalent, more like interesting examples of definitions.

Answer 1

Theorems tend to translate into definitions.

As people mentioned in the comments, in $\sf ZFC$ we can prove the theorem that a set is infinite if and only if it contains a countably infinite subset. In $\sf ZF$ this isn't true, so it becomes a type of definition.

In $\sf ZFC$ a ring is Noetherian if one of the two equivalent definitions hold:

1. Every collection of ideals contains a maximal element;
2. every increasing chain of ideals is finite.

In $\sf ZF$ the equivalent fails, and they turn into two separate definitions.

This extends to every theorem of $\sf ZFC$ that fails in $\sf ZF$.