3.7.E: Problems on Metric Spaces (Exercises)

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. $]$