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. \(]\)