Section 7: 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

Show that $\mathbb{Q}$ is countably infinite.

To show that $\mathbb{Q}$ is countably infinite, we can follow Example 3 on page 48 with the only difference that for $n,m\in\mathbb{Z}_{+}$ , $g(n,2m)=\frac{m-1}{n}$ , $g(n,2m-1)=-\frac{m}{n}$ .

Alternatively, $\mathbb{Q}=\mathbb{Q}_{+}\cup\{0\}\cup\mathbb{Q}_{-}$ , where the latter is in an obvious bijective correspondence with the former (so, we can use Theorem 7.5 and Example 3, page 48).

Subpages

Section 7: Problem 1 Solution

Section 7: Problem 1 Solution