Injection of union into disjoint union

Given a family of sets $(A_i : i \in I)$, we define the disjoint union: $$\sum_{i \in I} A_i = \bigcup_{i \in I} (\{i\} \times A_i).$$

There is a surjection $\sum_{i \in I}A_i \to \bigcup_{i \in I} A_i$ given by $(i,a) \mapsto a$, so by the Axiom of Choice there is an injection $\bigcup_{i \in I} A_i \to \sum_{i \in I}A_i$.

My question is whether AC (or some fragment of it) is required to prove that there is an injection $\bigcup_{i \in I} A_i \to \sum_{i \in I}A_i$.

(If I remember correctly, it is not known whether "every surjective image of any set $X$ injects into $X$" implies AC.)

Solution 1:

The assertion "If $f\colon X\to A$ is surjective then there is $g\colon A\to X$ injective" is known as The Partition Principle.

Indeed it is still open whether or not the partition principle implies the axiom of choice. (It is one of my goals for the foreseeable future, to solve this problem, although it's a goal I am fully aware I am unlikely to achieve.)

To see that the formulation $\bigcup A_i\leq\sum A_i$ is equivalent to the Partition Principle, one direction is trivial (clearly PP implies this thing), and in the other direction suppose $f\colon A\to B$ is surjective, take $I=A$ and $X_a=\{f(a)\}$. We have that $\bigcup X_a=B$ while $\sum X_a=f$ which is naturally in bijection with $A$. Requiring, if so, an injection from the union into the sum implies an injection from $B$ back into $A$, as wanted.

The assertion "every surjection splits" indeed implies the axiom of choice, that is to say the map $(i,a)\mapsto i$ is a surjection then there is an injection which splits it, is exactly asserting a choice function.

But requiring only the existence of at least one injection whenever there is a surjection is not enough to ensure every surjection splits, at least not as we know it. It goes even further, I don't think that we know that many counterexamples too.

The only sets I am aware of that have the property that whenever $f\colon X\to Y$ is surjective then there is $g\colon Y\to X$ injective are sets whose surjections split, and those are strong $\kappa$-amorphous sets, that is sets that every partition into two sets implies one is smaller than $\kappa$; and that every partition into no less than $\kappa$ parts is almost all singletons (almost all: meaning all but $<\kappa$ many parts).

For example, if $A$ is strongly amorphous (which means that it cannot be split into two infinite sets, and every infinite partition is all but finitely many singletons) then every surjection from $A$ splits.

I am not aware of any sets other than those which have the partition principle in ZF, or even consistently having partition properties.