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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 ( ho(overline{x}, overline{y})) are defined by
(a) ( ho(overline{x}, overline{y})=left|x_{1}-y_{1} ight|+left|x_{2}-y_{2} ight|) or
(b) ( ho(overline{x}, overline{y})=max left{left|x_{1}-y_{1} ight|,left|x_{2}-y_{2} ight| ight}),
where (overline{x}=left(x_{1}, x_{2} ight)) and (overline{y}=left(y_{1}, y_{2} ight) .) 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 ( ho) 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
[
ho(m, n)=left|frac{1}{m}-frac{1}{n} ight|, ext { with the convention that } frac{1}{infty}=0.
]
Verify the metric axioms. Describe (G_{infty}left(frac{1}{2} ight), S_{infty}left(frac{1}{2} ight),) and (G_{1}(1)).

Exercise (PageIndex{5})

(Rightarrow 5 .) Metrize the extended real number system (E^{*}) by
[
ho^{prime}(x, y)=|f(x)-f(y)|,
]
where the function
[
f : E^{*} underset{ ext { onto }}{longrightarrow}[-1,1]
]
is defined by
[
f(x)=frac{x}{1+|x|} ext { if } x ext { is finite, } f(-infty)=-1, ext { and } f(+infty)=1.
]
Compute ( ho^{prime}(0,+infty), ho^{prime}(0,-infty), ho^{prime}(-infty,+infty), ho^{prime}(0,1), ho^{prime}(1,2),) and ( ho^{prime}(n,+infty) .) Describe (G_{0}(1), G_{+infty}(1),) and (G_{-infty}left(frac{1}{2} ight) .) 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]
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 ( ho^{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

Exercise (PageIndex{7})

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

Exercise (PageIndex{8})

Metrize the closed interval ([0,+infty]) in (E^{*}) by setting
[
ho(x, y)=left|frac{1}{1+x}-frac{1}{1+y} ight| ,
]
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 ( ho) is a metric for (S,) then another metric ( ho^{prime}) for (S) is given by
(i) ( ho^{prime}(x, y)=min {1, ho(x, y)});
(ii) ( ho^{prime}(x, y)=frac{ ho(x, y)}{1+ ho(x, y)}).
In case ((mathrm{i}),) show that globes (G_{p}(varepsilon)) of radius (varepsilon leq 1) are the same under ( ho) and ( ho^{prime} .) In case (ii), prove that any (G_{p}(varepsilon)) in ((S, ho)) is also a globe (G_{p}left(varepsilon^{prime} ight)) in (left(S, ho^{prime} ight)) of radius
[
varepsilon^{prime}=frac{varepsilon}{1+varepsilon},
]
and any globe of radius (varepsilon^{prime}<1) in (left(S, ho^{prime} ight)) is also a globe in ((S, ho) .) (Find the converse formula for (varepsilon) as well!)
[Hint for the triangle inequality in (ii): Let (a= ho(x, z), b= ho(x, y),) and (c= ho(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, ho^{prime} ight)) and (left(Y, ho^{prime prime} ight)) are metric spaces, then a metric ( ho) for the set (X imes Y) is obtained by setting, for (x_{1}, x_{2} in X) and (y_{1}, y_{2} in Y),
(i) ( holeft(left(x_{1}, y_{1} ight),left(x_{2}, y_{2} ight) ight)=max left{ ho^{prime}left(x_{1}, x_{2} ight), ho^{prime prime}left(y_{1}, y_{2} ight) ight} ;) or
(ii) ( holeft(left(x_{1}, y_{1} ight),left(x_{2}, y_{2} ight) ight)=sqrt{ ho^{prime}left(x_{1}, x_{2} ight)^{2}+ ho^{prime prime}left(y_{1}, y_{2} ight)^{2}}).
[Hint: For brevity, put ( ho_{12}^{prime}= ho^{prime}left(x_{1}, x_{2} ight), ho_{12}^{prime prime}= ho^{prime prime}left(y_{1}, y_{2} ight),) etc. The triangle inequality in (ii),
[
sqrt{left( ho_{13}^{prime} ight)^{2}+left( ho_{13}^{prime prime} ight)^{2}} leq sqrt{left( ho_{12}^{prime} ight)^{2}+left( ho_{12}^{prime prime} ight)^{2}}+sqrt{left( ho_{23}^{prime} ight)^{2}+left( ho_{23}^{prime prime} ight)^{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.,
[
ho_{13}^{prime} leq ho_{12}^{prime}+ ho_{23}^{prime} ext { and } ho_{13}^{prime prime} leq ho_{12}^{prime prime}+ ho_{23}^{prime prime}.
]
Reverse all steps, so that the required inequality becomes the last step. (])

Exercise (PageIndex{11})

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

Exercise (PageIndex{12})

Prove that
[
holeft(p_{1}, p_{2} ight)+ holeft(p_{2}, p_{3} ight)+cdots+ holeft(p_{n-1}, p_{n} ight) geq holeft(p_{1}, p_{n} ight).
]
[Hint: Use induction. (])