3.8.E: Problems on Neighborhoods, Open and Closed Sets (Exercises)
\(\Rightarrow 1 .\) 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
\[
x \in G_{p}(\delta) \Rightarrow \rho(x, q)<r \Rightarrow x \in G_{q}(r) . ]
\]
\(\Rightarrow 2 .\) 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 \(G_{\overline{p}}(\delta) \subseteq(\overline{a}, \overline{b}) . ]\)
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 \(\overline{d} (\text { given } \overline{p}, \text { the center). Prove that }[\overline{c}, \overline{d}] \subseteq G_{\overline{p}}(\delta) .]\)
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 \(G_{\overline{p}}(\delta) \nsubseteq[\overline{a}, \overline{b}] . ]\)
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 \(G_{\overline{q}}(r)=\bigcup C_{p} . ]\)
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. \(]\)
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 \((1) . ]\)
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 \(k=1,2, \ldots, n .\)
Do Problem \(7,\) with \(G_{\overline{q}}(r)\) replaced by an arbitrary open set \(G \neq \emptyset\) in \(E^{n} .\)
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.]
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
\[
\bigcup_{n=2}^{\infty}\left[\frac{1}{n}, 1-\frac{1}{n}\right]=(0,1) . ]
\]
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\() . ]\)
Prove the last parts of Theorems 3 and 4.
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 \(A .\)
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 \(A^{0} \subseteq B^{0}\).
\(\left[\text { Hint for }(\mathrm{ii}) : A^{0} \text { is open by Problem } 14 .\right]\)
Is \(A^{0} \cup B^{0}=(A \cup B)^{0} ?\)
[Hint: See Example \((4) .\) Take \(A=R, B=E^{1}-R . ]\)
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 \(P \supseteq N\).
\([\text { Hint for }(\mathrm{c}) : \text { See Problem } 14 .]\)
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 \(R^{n}=E^{n}\left(R^{n} \text { as in Problem } 6\right)\).
Verify Example ( 8 ) for intervals in \(E^{n}\).