Section 21: Problem 12 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

Prove continuity of the algebraic operations on $\mathbb{R}$ , as follows: Use the metric $d(a,b)=|a-b|$ on $\mathbb{R}$ and the metric on $\mathbb{R}^{2}$ given by the equation $\rho((x,y),(x_{0},y_{0}))=\max\{|x-x_{0}|,|y-y_{0}|\}.$

(a) Show that addition is continuous. [Hint: Given $\epsilon$ , let $\delta=\epsilon/2$ and note that $d(x+y,x_{0}+y_{0})\le|x-x_{0}|+|y-y_{0}|\mbox{.]}$

(b) Show that multiplication is continuous. [Hint: Given $(x_{0},y_{0})$ and $0<\epsilon<1$ , let $3\delta=\epsilon/(|x_{0}|+|y_{0}|+1)$ and note that $d(xy,x_{0}y_{0})\le|x_{0}||y-y_{0}|+|y_{0}||x-x_{0}|+|x-x_{0}||y-y_{0}|\mbox{.]}$

(c) Show that the operation of taking reciprocals is a continuous map from $\mathbb{R}-\{0\}$ to $\mathbb{R}$ . [Hint: Show the inverse image of the interval $(a,b)$ is open. Consider five cases, according as $a$ and $b$ are positive, negative, or zero.]

(d) Show that the subtraction and quotient operations are continuous.

(a) Using the hint, if $(x,y)\in B_{\rho}((x_{0},y_{0}),\delta)$ then $d(x+y,x_{0}+y_{0})\le$ $|x-x_{0}|+|y-y_{0}|<2\delta=\epsilon$ , i.e. $x+y\in B_{d}(x_{0}+y_{0},\epsilon)$ .

(b) I am not sure why there is $3$ in the expression for $\delta$ . Also, instead of assuming $\epsilon<1$ , we can just take $\delta=\min\{\epsilon/(|x_{0}|+|y_{0}|+1),\tfrac{1}{2}\}$ . Using the hint, if $(x,y)\in B_{\rho}((x_{0},y_{0}),\delta)$ then $d(xy,x_{0}y_{0})\le$ $|x_{0}||y-y_{0}|+|y_{0}||x-x_{0}|+|x-x_{0}||y-y_{0}|\le$ $|x_{0}|\delta+|y_{0}|\delta+\delta^{2}<$ $|x_{0}|\delta+|y_{0}|\delta+\delta\le\epsilon$ , i.e. $xy\in B_{d}(x_{0}y_{0},\epsilon)$ .

(c) The inverse image of $(a,b)$ is (we may consider finite intervals only): $(1/b,1/a)$ if they have the same sign, $(-\infty,1/a)\cup(1/b,+\infty)$ if they have different signs, $(1/b,+\infty)$ if $a=0$ , and $(-\infty,1/a)$ if $b=0$ .

(d) Since $f(x)=-x$ is continuous (the preimage of any interval $(a,b)$ , $(-b,-a)$ , is open), by Exercise 10 of §18 as well as (a), (b) and (c), $x-y=x+(-y)$ and $\tfrac{x}{y}=x\cdot\tfrac{1}{y}$ are continuous as composites of continuous functions.

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Section 21: Problem 12 Solution

Section 21: Problem 12 Solution