Theorem 2.21

(CA) No closed interval is the union of two mutually exclusive point sets.

**Proof:**

Let $H=[a_{H},b_{H}]$ and $K=[a_{K},b_{K}]$ where $H$ and $K$ are mutually exclusive.

Since $H$ and $K$ are both closed point sets they each have a left most point and a right most point (by the definition of a closed point set).

If there is a point $p \in H$ between any two points $m$ and $n$ of $K$, then by axiom 1.6, there is a point $q$ between both ($m$ and $q$) and ($q$ and $n$). Thus $H \cup K$ is not a closed interval.

If $H<K$ then by axiom 1.6 there exists some point $p_x$between the right most point of $H$

and the left most point of $K$ where $p_x \notin H \cup K$ and again, no colosed interval exists.

By symetrical arguement for the case where $K<H$ there is a point between the the left most point and right most point of $K \cup H$ that is not a member.

$\therefore$ the union of two mutually exclusive closed point sets in not a closed interval.

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