Section 17: Problem 20 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

Find the boundary and the interior of each of the following subsets of $\mathbb{R}^{2}$ :

(a) $A=\{x\times y|y=0\}$

(b) $B=\{x\times y|x>0\mbox{ and }y\neq0\}$

(d) $D=\{x\times y|x\mbox{ is rational}\}$

(e) $E=\{x\times y|0<x^{2}-y^{2}\le1\}$

(f) $F=\{x\times y|x\neq0\mbox{ and }y\le1/x\}$

Set	Bd	+	Int	=	Cl
$A$	$A$		$\emptyset$		$A$
$B$	$\{0\}\times\mathbb{R}$ $\cup$ $\mathbb{R}_{+}\times\{0\}$		$B$		$\overline{\mathbb{R}}_{+}\times\mathbb{R}$
$C$	$\mathbb{R}_{-}\times\{0\}$ $\cup$ $\{0\}\times\mathbb{R}$		$\mathbb{R}_{+}\times\mathbb{R}$		$\mathbb{R}_{-}\times\{0\}$ $\cup$ $\overline{\mathbb{R}}_{+}\times\mathbb{R}$
$D$	$\mathbb{R}\times\mathbb{R}$		$\emptyset$		$\mathbb{R}\times\mathbb{R}$
$E$	$\{(x,y)\|\|x\|=\|y\|$ or $x^{2}-y^{2}=1\}$		$\{(x,y)\|0<x^{2}-y^{2}<1\}$		$\{(x,y)\|0\le x^{2}-y^{2}\le1\}$
$F$	$\{(x,y)\|(x\neq0$ and $y=1/x)$ or $x=0\}$		$\{(x,y)\|x\neq0$ and $y<1/x\}$		$\{(x,y)\|(x\neq0$ and $y\le1/x)$ or $x=0\}$

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Section 17: Problem 20 Solution

Section 17: Problem 20 Solution