Section 1: Problem 10 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

Let $\mathbb{R}$ denote the set of real numbers. For each of the following subsets of $\mathbb{R}\times\mathbb{R}$ , determine whether it is equal to the cartesian product of two subsets of $\mathbb{R}$ .

(a) $\{(x,y)|x\mbox{ is an integer}\}$ .

(b) $\{(x,y)|0<y\le1\}$ .

(d) $\{(x,y)|x\mbox{ is not an integer and }y\mbox{ is an integer}\}$ .

(e) $\{(x,y)|x^{2}+y^{2}<1\}$ .

A set $C$ in $\mathbb{R}\times\mathbb{R}$ is the cartesian product of two sets in $\mathbb{R}$ iff $\forall x\in\mathbb{R}$ the set $Y(x)$ of $y$ ’s such that $(x,y)\in C$ is either the empty set or does not depend on $x$ . So, we get the positive answer in (a), (b) and (d).

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Section 1: Problem 10 Solution

Section 1: Problem 10 Solution