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Let $A\ne\emptyset$. I want to prove that if $A$ is partitionable into denumerable sets then there exists an injective function from natural numbers to the set $A$. In the hint of this question it is said that one requires Zorn's lemma to prove it. However, I don't think that I'm doing it right. Here's my approach.

Since $A$ is partitionable into denumerable sets, there exists a collection $\{P_i\}_{i\in I}\subset \mathcal{P}(A)$ such that

  • each $P_i$ is denumerable;

  • $P_i \cap P_j=\emptyset$ if $i\ne j\in I$;

  • $A = \bigcup\limits_{i\in I} P_i$.

Since each $P_i$ is denumerable, there exists an injective function $f_i:\mathbb{N}\to P_i$.

Since the $P_i$ are denumerable, each $P_i$ can be partitioned into finite chains $P_{i_k}$. Then each chain will have an upper bound, and so $P_i$ has a maximal element, which also implies that $P_i$ is finite. Let $x_i \in P_i$ be a maximal element of $P_i$. Since $f_i$ is injective, $f^{-1}(x_i):=k_i\in\mathbb{N}$ exists.

Define

$$f(n) = \begin{cases} f_1 & n\in \{1,2,\dots, k_1\} \\ f_2 & n\in \{k_1+1,k_1+2,\dots, k_2\} \\ \vdots\\ f_m & n\in \{k_{m-1}+1,k_{m-1}+2,\dots, k_m\} \\ \vdots \end{cases} $$

Then $f:\mathbb{N}\to A$ is an injection, as required.

My questions are the following:

(1) Can we really say that each $P_i$ can be partitioned into finite chains $P_{i_k}$? I doubt this.

(2) Does there really necessarily exist a maximal element in $A$? If $A$ is partitionable into denumerable sets, this does not imply that $A$ is finite. Then $A$ does not necessarily have a maximal element.

(3) So how should Zorn's lemma be applied here exactly?

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closed as unclear what you're asking by Andrés E. Caicedo, user91500, JMP, B. Goddard, MAN-MADE Sep 18 '17 at 14:03

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  • $\begingroup$ Huh? I believe that $\mathbb N\times\mathbb R$ can be partitioned into denumerable sets $P_i=\mathbb N\times\{i\},\ i\in\mathbb R$ but I don't believe there is a surjection from $\mathbb N$ to $\mathbb N\times\mathbb R.$ $\endgroup$ – bof Sep 17 '17 at 21:49
  • $\begingroup$ Perhaps you mean to say that $A$ is partitionable into countably many countable sets? $\endgroup$ – Alex Provost Sep 17 '17 at 21:51
  • $\begingroup$ Sorry, my mistake, it should be "injective function" instead of "surjective function". Editing now. $\endgroup$ – sequence Sep 18 '17 at 0:41
  • $\begingroup$ Are denumerable sets infinite? Otherwise, the injective version of the question is false as well. $\endgroup$ – Andrés E. Caicedo Sep 18 '17 at 0:46
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    $\begingroup$ Sorry, I missed "$A\ne\emptyset.$" But the problem as stated now is trivial. If $A$ is partitioned into denumerable sets (and is nonempty), then $A$ contains a denumerable set $P.$ So there is an injective map $f:\mathbb N\to P\subseteq A,$ so $f$ is an injective map from $\mathbb N$ to $A.$ $\endgroup$ – bof Sep 18 '17 at 6:20
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The result is false. For each $t\in [0,1),$ define

$$A_t= \{n+t: n \in \mathbb Z\}.$$

Then each $A_t$ is denumerable, the sets $A_t$ are pairwise disjoint, and $\mathbb R = \bigcup_{t\in [0,1)}A_t.$

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  • $\begingroup$ Sorry, my mistake, it should be "injective function" instead of "surjective function". Editing now. $\endgroup$ – sequence Sep 18 '17 at 0:41
  • $\begingroup$ @sequence But you still say "It is in my understanding that what needs to be proved is that $A$ is also denumerable." That's clearly false. $\endgroup$ – zhw. Sep 18 '17 at 0:54
  • $\begingroup$ Yes, you're right, I think it's more clear to me that it doesn't have to be denumerable, since by definition a denumerable set $S$ is one which has a corresponding injection from $S$ to $\mathbb{N}$, and not vice versa. @zhw. $\endgroup$ – sequence Sep 18 '17 at 5:40

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