Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.

Question

Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.

Answer 1

First let us recall Zorn's lemma.

Zorn's lemma. Suppose that $(P,\leq)$ is a non-empty partial ordered set such that whenever $C\subseteq P$ is a chain, then there is $p\in P$ that for every $c\in C$, $c\leq p$. Then $(P,\leq)$ has a maximal element.

To use Zorn's lemma, if so, one has to find a partial order with the above property (every chain has an upper bound) and utilize the maximality to prove what is needed.

We shall use the partial order whose members are $(A,B)$ where $A,B$ are disjoint subsets of $\mathbb R^+$ each is closed under addition. We will say that $(A,B)\leq (A',B')$ if $A\subseteq A'$ and $B\subseteq B'$.

This is obviously a partial order. It is non-empty because we can take $A=\mathbb N\setminus\{0\}$ and $B=\{n\cdot\pi\mid n\in\mathbb N\setminus\{0\}\}$, both are clearly closed under addition and disjoint.

Suppose that $C=\{(A_i,B_i)\mid i\in I\}$ is a chain, let $A=\bigcup_{i\in I}A_i$ and $B=\bigcup_{i\in I} B_i$. To see that these sets are disjoint suppose $x\in A\cap B$ then for some $A_i$ and $B_j$ we have $x\in A_i\cap B_j$. Without loss of generality $i<j$ then $x\in A_j\cap B_j$ contradiction the assumption that $(A_j,B_j)\in P$ and therefore these are disjoint sets. The proof that $A$ and $B$ are closed under addition is similar.

Then $(A,B)\in P$ and therefore is an upper bound of $C$. So every chain has an upper bound and Zorn's lemma says that there is some $(X,Y)$ which is a maximal element.

Now all that is left is to show that $X\cup Y=\mathbb R^+$. Suppose that it wasn't then there was some $r\in\mathbb R^+$ which was neither in $X$ nor in $Y$, then we can take $X'$ to be the closure of $X\cup\{r\}$ under addition. If $X'\cap Y=\varnothing$ then $(X',Y)\in P$ and it is strictly above $(X,Y)$ which is a contradiction to the maximality. Therefore $X'\cap Y$ is non-empty, but then taking $Y'$ to be the closure of $Y\cup\{r\}$ under addition has to be disjoint from $X$, and the maximality argument holds again.

In either case we have that $X\cup Y=\mathbb R^+$.

Answer 2

Let $\mathcal{P}$ the set of the disjoint pairs $(A,B)$, where $A,B\subseteq\mathbb{R}^+$ are not empty and each one is closed under addition and multiplication by a positive rational number. Note that $\mathcal{P}\neq\emptyset$ because if we consider $X=\mathbb{Q}^+$ and $Y=\{n\sqrt{2}:n\in\mathbb{Q}^+\}$, then $(X,Y)\in\mathcal{P}$. Define $\leq$ as follows:

$(X_1,Y_1)\leq(X_2,Y_2)$ if and only if $X_1\subseteq X_2$ and $Y_1\subseteq Y_2$, for all $(X_1,Y_1),(X_2,Y_2)\in\mathcal{P}$.

Clearly, $(\mathcal{P},\leq)$ is a partially ordered set. Furthermore, it is easy to see that every chain in $(\mathcal{P},\leq)$ has an upper bound. We can now apply Zorn's lemma, so $(\mathcal{P},\leq)$ has a maximal element. Let $(A,B)\in\mathcal{P}$ a maximal element of $(\mathcal{P},\leq)$. We only have to show that $A\cup B=\mathbb{R}^+$.

Suppose that $\mathbb{R}^+\not\subseteq A\cup B$. Therefore, there exists $x\in\mathbb{R}^+$ such that $x\not\in A\cup B$. Consider $A_x=\{kx+a:k\in\mathbb{Q}^+\cup\{0\}\textrm{ and }a\in A\}$ and $B_x=\{kx+b:k\in\mathbb{Q}^+\cup\{0\}\textrm{ and }b\in B\}$. Then $A\subseteq A_x$, $B\subseteq B_x$ and $A_x,B_x\subseteq\mathbb{R}^+$. It is easy to verify that $A_x$ and $B_x$ are closed under addition and multiplication by a positive rational number.

If either $A_x\cap B=\emptyset$ or $A\cap B_x=\emptyset$, then $(A,B)$ would not be a maximal element of $(\mathcal{P},\leq)$. Therefore, $A_x\cap B\neq\emptyset$ and $A\cap B_x\neq\emptyset$, so there is some $q_0\in\mathbb{Q}^+$ and $a\in A$ such that $q_0x+a\in B$, and there is some $q_1\in\mathbb{Q}^+$ and $b\in B$ such that $q_1x+b\in A$. Note that $q_0,q_1\neq 0$ because $A\cap B=\emptyset$. It is easy to see that $q_0q_1x+q_1a+q_0b\in A\cap B$, so $A\cap B\neq\emptyset$, a contradiction. Therefore, $A\cup B=\mathbb{R}^+$.