Section 16: Problem 9 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 dictionary order topology on the set $\mathbb{R}\times\mathbb{R}$ is the same as the product topology $\mathbb{R}_{d}\times\mathbb{R}$ , where $\mathbb{R}_{d}$ denotes $\mathbb{R}$ in the discrete topology. Compare this topology with the standard topology on $\mathbb{R}^{2}$ .

Every interval in the dictionary order topology on $\mathbb{R}\times\mathbb{R}$ is the union of open sets in $\mathbb{R}_{d}\times\mathbb{R}$ . At the same time, a basis for $\mathbb{R}_{d}\times\mathbb{R}$ is the collection of sets $\{x\}\times(a,b)$ , where each is an interval in the dictionary order topology. Therefore, the topologies are the same.

These topologies are strictly finer then $\mathbb{R}^{2}$ , see Exercise 5. For example, $\{0\}\times\mathbb{R}$ is open in the product topology $\mathbb{R}_{d}\times\mathbb{R}$ (and, hence, in the dictionary topology on $\mathbb{R}\times\mathbb{R}$ ), but not in $\mathbb{R}^{2}$ .

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Section 16: Problem 9 Solution

Section 16: Problem 9 Solution