3.12.E: Problems on Cluster Points, Closed Sets, and Density

Exercise $\PageIndex{1}$

Complete the proof of Theorem 1$(\text { ii })$.

Exercise $\PageIndex{2}$

Prove that $\overline{R}=E^{1}$ and $\overline{R^{n}}=E^{n}(\text { Example }(\mathrm{a}))$.

Exercise $\PageIndex{3}$

Prove Theorem 2 for $E^{3} .$ Prove it for $E^{n}\left(^{*} \text { and } C^{n}\right)$ by induction on $n .$

Exercise $\PageIndex{4}$

Verify Note 2.

Exercise $\PageIndex{5}$

Prove Theorem 3.

Exercise $\PageIndex{6}$

Prove Corollaries 1 and 2.

Exercise $\PageIndex{7}$

Prove that $(A \cup B)^{\prime}=A^{\prime} \cup B^{\prime}$.
[Hint: Show by contradiction that $p \notin\left(A^{\prime} \cup B^{\prime}\right)$ excludes $p \in(A \cup B)^{\prime} .$ Hence $(A \cup B)^{\prime} \subseteq A^{\prime} \cup B^{\prime} .$ Then show that $A^{\prime} \subseteq(A \cup B)^{\prime},$ etc. $]$

Exercise $\PageIndex{8}$

From Problem $7,$ deduce that $A \cup B$ is closed if $A$ and $B$ are. Then prove Corollary $4 .$ By induction, extend both assertions to any finite number of sets.

Exercise $\PageIndex{9}$

From Theorem $4,$ prove that if the sets $A_{i}(i \in I)$ are closed, so is $\bigcap_{i \in I} A_{i}$.

Exercise $\PageIndex{10}$

Prove Corollary 3 from Theorem 3. Deduce that $\overline{\overline{A}}=\overline{A}$ and prove footnote $3 .$
[Hint: Consider Figure 7 and Example $(1)$ in §12 when using Theorem 3 (twice). $]$

Exercise $\PageIndex{11}$

Prove that $\overline{A}$ is contained in any closed superset of $A$ and is the intersection of all such supersets.
[Hint: Use Corollaries 2 and $3 . ]$

Exercise $\PageIndex{12}$

(i) Prove that a bounded sequence $\left\{\overline{x}_{m}\right\} \subseteq E^{n}\left(^{*} C^{n}\right)$ converges to $\overline{p}$ iff $\overline{p}$ is its only cluster point.
(ii) Disprove it for
(a) unbounded $\left\{\overline{x}_{m}\right\}$ and
(b) other spaces.
[Hint: For $(\mathrm{i}),$ if $\overline{x}_{m} \rightarrow \overline{p}$ fails, some $G_{\overline{p}}$ leaves out infinitely many $\overline{x}_{m} .$ These $\overline{x}_{m}$ form a bounded subsequence that, by Theorem $2,$ clusters at some $\overline{q} \neq \overline{p} .$ (Why? $)$ Thus $\overline{q}$ is another cluster point (contradiction!)
For (ii), consider (a) Example (f) in §14 and (b) Problem 10 in §14, with (0,2] as a subspace of $E^{1} . ]$

Exercise $\PageIndex{13}$

In each case of Problem 10 in §14, find $\overline{A}$. Is $A$ closed? (Use Theorem 4.)

Exercise $\PageIndex{14}$

Prove that if $\left\{b_{n}\right\} \subseteq B \subseteq \overline{A}$ in $(S, \rho),$ there is a sequence $\left\{a_{n}\right\} \subseteq A$ such that $\rho\left(a_{n}, b_{n}\right) \rightarrow 0 .$ Hence $a_{n} \rightarrow p$ iff $b_{n} \rightarrow p .$
[Hint: Choose $a_{n} \in G_{b_{n}}(1 / n) .]$

Exercise $\PageIndex{15}$

We have, by definition,
$$p \in A^{0} \text { iff }(\exists \delta>0) G_{p}(\delta) \subseteq A;$$
hence
$$p \notin A^{0} \text { iff }(\forall \delta>0) G_{p}(\delta) \nsubseteq A, \text { i.e., } G_{p}(\delta)-A \neq \emptyset .$$
(See Chapter $1,§§1-3 . )$ Find such quantifier formulas for $p \in \overline{A}, p \notin \overline{A}$, $p \in A^{\prime},$ and $p \notin A^{\prime}$.
[Hint: Use Corollary 6 in $§ 14,$ and Theorem 3 in $§16 .]$

Exercise $\PageIndex{16}$

Use Problem 15 to prove that
(i) $-(\overline{A})=(-A)^{0}$ and
(ii) $-\left(A^{0}\right)=\overline{-A}$.

Exercise $\PageIndex{17}$

Show that
$$\overline{A} \cap(\overline{-A})=\mathrm{bd} A(\text { boundary of } A);$$
cf. $§ 12,$ Problem $18 .$ Hence prove again that $A$ is closed iff $A \supseteq$ bd $A .$
[Hint: Use Theorem 4 and Problem 16 above. $]$

Exercise $\PageIndex{*18}$

A set $A$ is said to be nowhere dense in $(S, \rho)$ iff $(\overline{A})^{0}=\emptyset .$ Show that Cantor's set $P(§14, \text { Problem } 17)$ is nowhere dense.
$[\text { Hint: } P \text { is closed, so } \overline{P}=P .]$

Exercise $\PageIndex{*19}$

Give another proof of Theorem 2 for $E^{1}$.
[Hint: Let $A \subseteq[a, b] .$ Put
$$Q=\{x \in[a, b] | x \text { exceeds infinitely many points (or terms) of } A\}.$$
Show that $Q$ is bounded and nonempty, so it has a glb, say, $p=\inf A .$ Show that $A$ clusters at $p . ]$

Exercise $\PageIndex{*20}$

For any set $A \subseteq(S, \rho)$ define
$$G_{A}(\varepsilon)=\bigcup_{x \in A} G_{x}(\varepsilon).$$
Prove that
$$\overline{A}=\bigcap_{n=1}^{\infty} G_{A}\left(\frac{1}{n}\right).$$

Exercise $\PageIndex{*21}$

Prove that
$$\overline{A}=\{x \in S | \rho(x, A)=0\} ; \text { see } \$ 13, \text { Note } 3.$$
Hence deduce that a set $A$ in $(S, \rho)$ is closed iff
$(\forall x \in S) \quad \rho(x, A)=0 \Longrightarrow x \in A$.