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

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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 $$E^{n}$$.