$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.7.E: Problems on Metric Spaces (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

The "arrowed" problems should be noted for later work.

Exercise $$\PageIndex{1}$$

Show that $$E^{2}$$ becomes a metric space if distances $$\rho(\overline{x}, \overline{y})$$ are defined by
(a) $$\rho(\overline{x}, \overline{y})=\left|x_{1}-y_{1}\right|+\left|x_{2}-y_{2}\right|$$ or
(b) $$\rho(\overline{x}, \overline{y})=\max \left\{\left|x_{1}-y_{1}\right|,\left|x_{2}-y_{2}\right|\right\}$$,
where $$\overline{x}=\left(x_{1}, x_{2}\right)$$ and $$\overline{y}=\left(y_{1}, y_{2}\right) .$$ In each case, describe $$G_{\overline{0}}(1)$$ and $$S_{\overline{0}}(1) .$$ Do the same for the subspace of points with nonnegative coordinates.

Exercise $$\PageIndex{2}$$

Prove the assertions made in the text about globes in a discrete space. Find an empty sphere in such a space. Can a sphere contain the entire space?

Exercise $$\PageIndex{3}$$

Show that $$\rho$$ in Examples $$(3)$$ and $$(5)$$ obeys the metric axioms.

Exercise $$\PageIndex{4}$$

Let $$M$$ be the set of all positive integers together with the "point" $$\infty .$$ Metrize $$M$$ by setting
$\rho(m, n)=\left|\frac{1}{m}-\frac{1}{n}\right|, \text { with the convention that } \frac{1}{\infty}=0.$
Verify the metric axioms. Describe $$G_{\infty}\left(\frac{1}{2}\right), S_{\infty}\left(\frac{1}{2}\right),$$ and $$G_{1}(1)$$.

Exercise $$\PageIndex{5}$$

$$\Rightarrow 5 .$$ Metrize the extended real number system $$E^{*}$$ by
$\rho^{\prime}(x, y)=|f(x)-f(y)|,$
where the function
$f : E^{*} \underset{\text { onto }}{\longrightarrow}[-1,1]$
is defined by
$f(x)=\frac{x}{1+|x|} \text { if } x \text { is finite, } f(-\infty)=-1, \text { and } f(+\infty)=1.$
Compute $$\rho^{\prime}(0,+\infty), \rho^{\prime}(0,-\infty), \rho^{\prime}(-\infty,+\infty), \rho^{\prime}(0,1), \rho^{\prime}(1,2),$$ and $$\rho^{\prime}(n,+\infty) .$$ Describe $$G_{0}(1), G_{+\infty}(1),$$ and $$G_{-\infty}\left(\frac{1}{2}\right) .$$ Verify the metric axioms (also when infinities are involved).

Exercise $$\PageIndex{6}$$

$$\Rightarrow 6 .$$ In Problem $$5,$$ show that the function $$f$$ is one to one, onto $$[-1,1],$$ and increasing; i.e.
$x<x^{\prime} \text { implies } f(x)<f\left(x^{\prime}\right) \text { for } x, x^{\prime} \in E^{*}.$
Also show that the $$f$$ -image of an interval $$(a, b) \subseteq E^{*}$$ is the interval $$(f(a), f(b)) .$$ Hence deduce that globes in $$E^{*}$$ (with $$\rho^{\prime}$$ as in Problem 5) are intervals in $$E^{*}$$ (possibly infinite).
[Hint: For a finite $$x,$$ put
$y=f(x)=\frac{x}{1+|x|}.$
Solving for $$x$$ (separately in the cases $$x \geq 0$$ and $$x<0 ),$$ show that
$(\forall y \in(-1,1)) \quad x=f^{-1}(y)=\frac{y}{1-|y|};$
thus $$x$$ is uniquely determined by $$y,$$ i.e., $$f$$ is one to one and onto-each $$y \in(-1,1)$$ corresponds to some $$x \in E^{1} .$$ (How about $$\pm 1 ? )$$
To show that $$f$$ is increasing, consider separately the three cases $$x<0<x^{\prime}$$, $$x<x^{\prime}<0$$ and $$0<x<x^{\prime}$$ (also for infinite $$x$$ and $$x^{\prime} ) . ]$$

Exercise $$\PageIndex{7}$$

Continuing Problems 5 and $$6,$$ consider $$\left(E^{1}, \rho^{\prime}\right)$$ as a subspace of $$\left(E^{*}, \rho^{\prime}\right)$$ with $$\rho^{\prime}$$ as in Problem $$5 .$$ Show that globes in $$\left(E^{1}, \rho^{\prime}\right)$$ are exactly all open intervals in $$E^{*} .$$ For example, $$(0,1)$$ is a globe. What are its center and radius under $$\rho^{\prime}$$ and under the standard metric $$\rho ?$$

Exercise $$\PageIndex{8}$$

Metrize the closed interval $$[0,+\infty]$$ in $$E^{*}$$ by setting
$\rho(x, y)=\left|\frac{1}{1+x}-\frac{1}{1+y}\right| ,$
with the conventions $$1+(+\infty)=+\infty$$ and $$1 /(+\infty)=0 .$$ Verify the metric axioms. Describe $$G_{p}(1)$$ for arbitrary $$p \geq 0$$.

Exercise $$\PageIndex{9}$$

Prove that if $$\rho$$ is a metric for $$S,$$ then another metric $$\rho^{\prime}$$ for $$S$$ is given by
(i) $$\rho^{\prime}(x, y)=\min \{1, \rho(x, y)\}$$;
(ii) $$\rho^{\prime}(x, y)=\frac{\rho(x, y)}{1+\rho(x, y)}$$.
In case $$(\mathrm{i}),$$ show that globes $$G_{p}(\varepsilon)$$ of radius $$\varepsilon \leq 1$$ are the same under $$\rho$$ and $$\rho^{\prime} .$$ In case (ii), prove that any $$G_{p}(\varepsilon)$$ in $$(S, \rho)$$ is also a globe $$G_{p}\left(\varepsilon^{\prime}\right)$$ in $$\left(S, \rho^{\prime}\right)$$ of radius
$\varepsilon^{\prime}=\frac{\varepsilon}{1+\varepsilon},$
and any globe of radius $$\varepsilon^{\prime}<1$$ in $$\left(S, \rho^{\prime}\right)$$ is also a globe in $$(S, \rho) .$$ (Find the converse formula for $$\varepsilon$$ as well!)
[Hint for the triangle inequality in (ii): Let $$a=\rho(x, z), b=\rho(x, y),$$ and $$c=\rho(y, z)$$ so that $$a \leq b+c .$$ The required inequality is
$\frac{a}{1+a} \leq \frac{b}{1+b}+\frac{c}{1+c}.$
Simplify it and show that it follows from $$a \leq b+c . ]$$

Exercise $$\PageIndex{10}$$

Prove that if $$\left(X, \rho^{\prime}\right)$$ and $$\left(Y, \rho^{\prime \prime}\right)$$ are metric spaces, then a metric $$\rho$$ for the set $$X \times Y$$ is obtained by setting, for $$x_{1}, x_{2} \in X$$ and $$y_{1}, y_{2} \in Y$$,
(i) $$\rho\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\max \left\{\rho^{\prime}\left(x_{1}, x_{2}\right), \rho^{\prime \prime}\left(y_{1}, y_{2}\right)\right\} ;$$ or
(ii) $$\rho\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right)=\sqrt{\rho^{\prime}\left(x_{1}, x_{2}\right)^{2}+\rho^{\prime \prime}\left(y_{1}, y_{2}\right)^{2}}$$.
[Hint: For brevity, put $$\rho_{12}^{\prime}=\rho^{\prime}\left(x_{1}, x_{2}\right), \rho_{12}^{\prime \prime}=\rho^{\prime \prime}\left(y_{1}, y_{2}\right),$$ etc. The triangle inequality in (ii),
$\sqrt{\left(\rho_{13}^{\prime}\right)^{2}+\left(\rho_{13}^{\prime \prime}\right)^{2}} \leq \sqrt{\left(\rho_{12}^{\prime}\right)^{2}+\left(\rho_{12}^{\prime \prime}\right)^{2}}+\sqrt{\left(\rho_{23}^{\prime}\right)^{2}+\left(\rho_{23}^{\prime \prime}\right)^{2}},$
is verified by squaring both sides, isolating the remaining square root on the right side, simplifying, and squaring again. Simplify by using the triangle inequalities valid in $$X$$ and $$Y,$$ i.e.,
$\rho_{13}^{\prime} \leq \rho_{12}^{\prime}+\rho_{23}^{\prime} \text { and } \rho_{13}^{\prime \prime} \leq \rho_{12}^{\prime \prime}+\rho_{23}^{\prime \prime}.$
Reverse all steps, so that the required inequality becomes the last step. $$]$$

Exercise $$\PageIndex{11}$$

Prove that
$|\rho(y, z)-\rho(x, z)| \leq \rho(x, y)$
in any metric space $$(S, \rho) .$$
[Caution: The formula $$\rho(x, y)=|x-y|,$$ valid in $$E^{n},$$ cannot be used in $$(S, \rho) .$$ Why? $$]$$

Exercise $$\PageIndex{12}$$

Prove that
$\rho\left(p_{1}, p_{2}\right)+\rho\left(p_{2}, p_{3}\right)+\cdots+\rho\left(p_{n-1}, p_{n}\right) \geq \rho\left(p_{1}, p_{n}\right).$
[Hint: Use induction. $$]$$