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

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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
$$(\forall x \in S) \quad \rho(x, A)=0 \Longrightarrow x \in A$$.