3.8.E: Problems on Neighborhoods, Open and Closed Sets (Exercises)

Exercise $\PageIndex{1}$

$\Rightarrow 1 .$ Verify Example $(1)$.
$\left[\text { Hint: Given } p \in G_{q}(r), \text { let }\right.$
$$\delta=r-\rho(p, q)>0 . \quad(\mathrm{Why}>0 ?)$$
Use the triangle law to show that
$$x \in G_{p}(\delta) \Rightarrow \rho(x, q)<r \Rightarrow x \in G_{q}(r) . ]$$

Exercise $\PageIndex{2}$

$\Rightarrow 2 .$ Check Example $(2) ;$ see Figure 8.
[Hint: If $\overline{p} \in(\overline{a}, \overline{b}),$ choose $\delta$ less than the 2$n$ numbers
$$p_{k}-a_{k} \text { and } b_{k}-p_{k}, \quad k=1, \ldots, n;$$
then show that $G_{\overline{p}}(\delta) \subseteq(\overline{a}, \overline{b}) . ]$

Exercise $\PageIndex{3}$

Prove that if $\overline{p} \in G_{\overline{q}}(r)$ in $E^{n},$ then $G_{\overline{q}}(r)$ contains a cube $[\overline{c}, \overline{d}]$ with $\overline{c} \neq \overline{d}$ and with center $\overline{p}$.
[Hint: By Example $(1),$ there is $G_{\overline{p}}(\delta) \subseteq G_{\overline{q}}(r) .$ Inscribe in $G_{\overline{p}}\left(\frac{1}{2} \delta\right)$ a cube of diagonal $\delta .$ Find its edge-length $(\delta / \sqrt{n}) .$ Then use it to find the coordinates of the endpoints, $\overline{c}$ and $\overline{d} (\text { given } \overline{p}, \text { the center). Prove that }[\overline{c}, \overline{d}] \subseteq G_{\overline{p}}(\delta) .]$

Exercise $\PageIndex{4}$

Verify Example $(3)$.
[Hint: To show that no interior points of $[\overline{a}, \overline{b}]$ are outside $(\overline{a}, \overline{b}),$ let $\overline{p} \notin(\overline{a}, \overline{b}) .$ Then at least one of the inequalities $a_{k}<p_{k}$ or $p_{k}<b_{k}$ fails. (Why?) Let it be $a_{1}<p_{1},$ say, so $p_{1} \leq a_{1}$.
Now take any globe $G_{\overline{p}}(\delta)$ about $\overline{p}$ and prove that it is not contained in $[\overline{a}, \overline{b}]$ (so $\overline{p}$ cannot be an interior point). For this purpose, as in Problem $3,$ show that $G_{\overline{p}}(\delta) \supseteq[\overline{c}, \overline{d}]$ with $c_{1}<p_{1} \leq a_{1} .$ Deduce that $\overline{c} \in G_{\overline{p}}(\delta),$ but $\overline{c} \notin[\overline{a}, \overline{b}] ;$ so $G_{\overline{p}}(\delta) \nsubseteq[\overline{a}, \overline{b}] . ]$

Exercise $\PageIndex{5}$

Prove that each open globe $G_{\overline{q}}(r)$ in $E^{n}$ is a union of cubes (which can be made open, closed, half-open, etc., as desired). Also, show that each open interval $(\overline{a}, \overline{b}) \neq \emptyset$ in $E^{n}$ is a union of open (or closed) globes.
[Hint for the first part: By Problem $3,$ each $\overline{p} \in G_{\overline{q}}(r)$ is in a cube $C_{p} \subseteq G_{\overline{q}}(r) .$ Show that $G_{\overline{q}}(r)=\bigcup C_{p} . ]$

Exercise $\PageIndex{6}$

Show that every globe in $E^{n}$ contains rational points, i.e., those with rational coordinates only (we express it by saying that the set $R^{n}$ of such points is dense in $E^{n} ) ;$ similarly for the set $I^{n}$ of irrational points (those with irrational coordinates).
[Hint: First check it with globes replaced by cubes $(\overline{c}, \overline{d}) ;$ see $§7,$ Corollary $3 .$ Then use Problem 3 above. $]$

Exercise $\PageIndex{7}$

Prove that if $\overline{x} \in G_{\overline{q}}(r)$ in $E^{n},$ there is a rational point $\overline{p}$ (Problem 6$)$ and a rational number $\delta>0$ such that $\overline{x} \in G_{\overline{p}}(\delta) \subseteq G_{\overline{q}}(r) .$ Deduce that each globe $G_{\overline{q}}(r)$ in $E^{n}$ is a union of rational globes (those with rational centers and radii). Similarly, show that $G_{\overline{q}}(r)$ is a union of intervals with rational endpoints.
[Hint for the first part: Use Problem 6 and Example $(1) . ]$

Exercise $\PageIndex{8}$

Prove that if the points $p_{1}, \ldots, p_{n}$ in $(S, \rho)$ are distinct, there is an $\varepsilon>0$ such that the globes $G\left(p_{k} ; \varepsilon\right)$ are disjoint from each other, for $k=1,2, \ldots, n .$

Exercise $\PageIndex{9}$

Do Problem $7,$ with $G_{\overline{q}}(r)$ replaced by an arbitrary open set $G \neq \emptyset$ in $E^{n} .$

Exercise $\PageIndex{10}$

Show that every open set $G \neq \emptyset$ in $E^{n}$ is infinite $(* \text { even uncountable; }$ see Chapter $1,§9$ ).
[Hint: Choose $G_{\overline{q}}(r) \subseteq G .$ By Problem $3, G_{\overline{p}}(r) \supset L[\overline{c}, \overline{d}],$ a line segment.]

Exercise $\PageIndex{11}$

Give examples to show that an infinite intersection of open sets may not be open, and an infinite union of closed sets may not be closed.
[Hint: Show that
$$\bigcap_{n=1}^{\infty}\left(-\frac{1}{n}, \frac{1}{n}\right)=\{0\}$$
and
$$\bigcup_{n=2}^{\infty}\left[\frac{1}{n}, 1-\frac{1}{n}\right]=(0,1) . ]$$

Exercise $\PageIndex{12}$

Verify Example $(6)$ as suggested in Figures 9 and $10 .$
[Hints: (i) For $\overline{G}_{q}(r),$ take
$$\delta=\rho(p, q)-r>0. \quad(\mathrm{Why}>0 ?)$$
(ii) If $\overline{p} \notin[\overline{a}, \overline{b}],$ at least one of the 2$n$ inequalities $a_{k} \leq p_{k}$ or $p_{k} \leq b_{k}$ fails (why?), say, $p_{1}<a_{1} .$ Take $\delta=a_{1}-p_{1}$.
In both $(\mathrm{i})$ and (ii) prove that $A \cap G_{p}(\delta)=\emptyset$ (proceed as in Theorem 1$) . ]$

Exercise $\PageIndex{*13}$

Prove the last parts of Theorems 3 and 4.

Exercise $\PageIndex{*14}$

Prove that $A^{0},$ the interior of $A,$ is the union of all open globes contained in $A$ (assume $A^{0} \neq \emptyset ) .$ Deduce that $A^{0}$ is an open set, the largest contained in $A .$

Exercise $\PageIndex{*15}$

For sets $A, B \subseteq(S, \rho),$ prove that
(i) $(A \cap B)^{0}=A^{0} \cap B^{0}$;
(ii) $\left(A^{0}\right)^{0}=A^{0} ;$ and
(iii) if $A \subseteq B$ then $A^{0} \subseteq B^{0}$.
$\left[\text { Hint for }(\mathrm{ii}) : A^{0} \text { is open by Problem } 14 .\right]$

Exercise $\PageIndex{16}$

Is $A^{0} \cup B^{0}=(A \cup B)^{0} ?$
[Hint: See Example $(4) .$ Take $A=R, B=E^{1}-R . ]$

Exercise $\PageIndex{17}$

Prove that if $M$ and $N$ are neighborhoods of $p$ in $(S, \rho),$ then
(a) $p \in M \cap N$;
(b) $M \cap N$ is a neighborhood of $p$;
*(c) so is $M^{0} ;$ and
(d) so also is each set $P \subseteq S$ such that $P \supseteq M$ or $P \supseteq N$.
$[\text { Hint for }(\mathrm{c}) : \text { See Problem } 14 .]$

Exercise $\PageIndex{18}$

The boundary of a set $A \subseteq(S, \rho)$ is defined by
$$\operatorname{bd} A=-\left[A^{0} \cup(-A)^{0}\right];$$
thus it consists of points that fail to be interior in $A$ or in $-A$.
Prove that the following statements are true:
(i) $S=A^{0} \cup$ bd $A \cup(-A)^{0},$ all disjoint.
(ii) $\operatorname{bd} S=\emptyset,$ bd $\emptyset=\emptyset$.
$*(\text { iii }) A$ is open iff $A \cap$ bd $A=\emptyset ; A$ is closed iff $A \supseteq$ bd $A$.
$(\mathrm{iv}) \operatorname{In} E^{n}$,
$$\operatorname{bd} G_{\overline{p}}(r)=\operatorname{bd} \overline{G}_{\overline{p}}(r)=S_{\overline{p}}(r)$$
(the sphere with center $\overline{p}$ and radius $r ) .$ Is this true in all metric spaces?
[Hint: Consider $G_{p}(1)$ in a discrete space $(S, \rho)$ with more than one point in $S ;$ see §11, Example (3).]
$(\mathrm{v}) \operatorname{In} E^{n},$ if $(\overline{a}, \overline{b}) \neq \emptyset,$ then
$\operatorname{bd}(\overline{a}, \overline{b}]=\operatorname{bd}[\overline{a}, \overline{b})=\operatorname{bd}(\overline{a}, \overline{b})=\operatorname{bd}[\overline{a}, \overline{b}]=[\overline{a}, \overline{b}]-(\overline{a}, \overline{b})$.
(vi) $\operatorname{In} E^{n},\left(R^{n}\right)^{0}=\emptyset ;$ hence bd $R^{n}=E^{n}\left(R^{n} \text { as in Problem } 6\right)$.

Exercise $\PageIndex{19}$

Verify Example ( 8 ) for intervals in $E^{n}$.