3.12.E: Problems on Cluster Points, Closed Sets, and Density
Complete the proof of Theorem 1\((\text { ii })\).
Prove that \(\overline{R}=E^{1}\) and \(\overline{R^{n}}=E^{n}(\text { Example }(\mathrm{a}))\).
Prove Theorem 2 for \(E^{3} .\) Prove it for \(E^{n}\left(^{*} \text { and } C^{n}\right)\) by induction on \(n .\)
Verify Note 2.
Prove Theorem 3.
Prove Corollaries 1 and 2.
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. \(]\)
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.
From Theorem \(4,\) prove that if the sets \(A_{i}(i \in I)\) are closed, so is \(\bigcap_{i \in I} A_{i}\).
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). \(]\)
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 \(3 . ]\)
(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 \(E^{1} . ]\)
In each case of Problem 10 in §14, find \(\overline{A}\). Is \(A\) closed? (Use Theorem 4.)
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 \(a_{n} \in G_{b_{n}}(1 / n) .]\)
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 \(§16 .]\)
Use Problem 15 to prove that
(i) \(-(\overline{A})=(-A)^{0}\) and
(ii) \(-\left(A^{0}\right)=\overline{-A}\).
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. \(]\)
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.
\([\text { Hint: } P \text { is closed, so } \overline{P}=P .]\)
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 \(p . ]\)
For any set \(A \subseteq(S, \rho)\) define
\[
G_{A}(\varepsilon)=\bigcup_{x \in A} G_{x}(\varepsilon).
\]
Prove that
\[
\overline{A}=\bigcap_{n=1}^{\infty} G_{A}\left(\frac{1}{n}\right).
\]
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
\((\forall x \in S) \quad \rho(x, A)=0 \Longrightarrow x \in A\).