Section 16: Problem 6 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 the countable collection $\{(a,b)\times(c,d)|a<b\mbox{ and }c<d,\mbox{ and }a,b,c,d\mbox{ are rational}\}$ is a basis for $\mathbb{R}^{2}$ .

The collection of sets $(a,b)$ such that $a,b\in\mathbb{Q}$ is a basis for the standard topology on $\mathbb{R}$ (see Exercise 8(a) of §13).

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Section 16: Problem 6 Solution

Section 16: Problem 6 Solution