Section 11*: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

If $a$ and $b$ are real numbers, define $a<b$ if $b-a$ is positive and rational. Show this is a strict partial order on $\mathbb{R}$ . What are the maximal simply ordered subsets?

It is nonreflexive ( $a-a\not>0$ ) and transitive ( $b-a,c-b\in\mathbb{Q}_{+}$ implies $c-a\in\mathbb{Q}_{+}$ ). The maximal subsets are $a+\mathbb{Q}$ for $a\in\mathbb{R}$ .

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Section 11*: Problem 1 Solution

Section 11*: Problem 1 Solution