Talk:Surjection iff Right Inverse

The statement that any surjection has a right inverse is in fact equivalent to the axiom of choice. Is there a page about this? I can easily provide three equivalent forms of the axiom of choice, and a proof of equivalence:


 * Any surjection $f: X \to Y$ has a right inverse.


 * For every $X\neq \emptyset$ there exists a function:
 * ${\displaystyle f: \mathcal{P}\left(X\right)\setminus\left\{\emptyset\right\} \to X}$

such that $f(Y)\in Y$ for each $Y\subset X$ with $Y\neq \emptyset$.


 * Let $\left(X_i\right)_{i\in I}$ be a family of sets such that $X_i \neq \emptyset$ for each $i\in I$.
 * Then:
 * ${\displaystyle \prod_{i\in I} X_i \neq \emptyset}$,
 * where
 * $\prod_{i\in I} X_i = \left\{ f: I \to \bigcup_{i\in I} X_i : f(i) \in X_i \ \forall i\in I\right\}. $

Who has ideas on how to add them? I do not want to add it while there are other proofs of equivalence on, say, two out of these.

Second of all, I have found plenty more equivalent forms of the axiom of choice, while these three easily follow from on another, the others might not... JSchoone 16:31, 31 January 2012 (EST)