3.7.E: Problems on Metric Spaces (Exercises)
- Page ID
- 22265
The "arrowed" problems should be noted for later work.
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.
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?
Show that \(\rho\) in Examples \((3)\) and \((5)\) obeys the metric axioms.
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 \(G_{1}(1)\).
\(\Rightarrow 5 .\) 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).
\(\Rightarrow 6 .\) 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 \(x^{\prime} ) . ]\)
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 \(\rho ?\)
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 \(p \geq 0\).
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 \(a \leq b+c . ]\)
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. \(]\)
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? \(]\)
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. \(]\)