4.10.E: Problems on Arcs, Curves, and Connected Sets
Discuss Examples \((a)\) and \((b)\) in detail. In particular, verify that \(L[\overline{a}, \overline{b}]\) is a simple arc. (Show that the map \(f\) in Example \((1)\) of §8 is one to one.)
Show that each polygon
\[
K=\bigcup_{i=0}^{m-1} L\left[\overline{p}_{i}, \overline{p}_{i+1}\right]
\]
can be reduced to a simple polygon \(P(P \subseteq K)\) joining \(p_{0}\) and \(p_{m}\).
[Hint: First, show that if two line segments have two or more common points, they lie in one line. Then use induction on the number \(m\) of segments in \(K .\) Draw a diagram in \(E^{2}\) as a guide.
Prove Theorem 1 of §9 for an arcwise connected \(B \subseteq(S, \rho)\).
[Hint: Proceed as in Problems 4 and 5 in §9, replacing \(g\) by some continuous map \(f :[a, b] \longrightarrow B .]\)
Define \(f\) as in Example \((\mathrm{f})\) of §1. Let
\[
G_{a b}=\left\{(x, y) \in E^{2} | a \leq x \leq b, y=f(x)\right\} .
\]
\(\left(G_{a b} \text { is the graph of } f \text { over }[a, b] .\right)\) Prove the following:
(i) If \(a>0,\) then \(G_{a b}\) is a simple arc in \(E^{2}\).
(ii) If \(a \leq 0 \leq b, G_{a b}\) is not even arcwise connected.
[Hints: (i) Prove that \(f\) is continuous on \([a, b], a>0,\) using the continuity of the
sine function. Then use Problem 16 in §2, restricting \(f\) to \([a, b] .\)
\(\left.\text { (ii) For a contradiction, assume } \overline{0} \text { is joined by a simple arc to some } \overline{p} \in G_{a b} .\right]\)
Show that each arc is a continuous image of \([0,1]\).
[ Hint: First, show that any \([a, b] \subseteq E^{1}\) is such an image. Then use a suitable composite mapping.
Prove that a function \(f : B \rightarrow E^{1}\) on a compact set \(B \subseteq E^{1}\) must be continuous if its graph,
\[
\left\{(x, y) \in E^{2} | x \in B, y=f(x)\right\} ,
\]
is a compact set (e.g., an arc) in \(E^{2}\).
[Hint: Proceed as in the proof of Theorem 3 of §8.]
Prove that \(A\) is connected iff there is no continuous map
\[
f : A \underset{\text { onto }}{\longrightarrow}\{0,1\} .
\]
[Hint: If there is such a map, Theorem 1 shows that \(A\) is disconnected. (Why?)
Conversely, if \(A=P \cup Q(P, Q \text { as in Definition } 3),\) put \(f=0\) on \(P\) and \(f=1\) on \(Q\). Use again Theorem 1 to show that \(f\) so defined is continuous on \(A\).]
Let \(B \subseteq A \subseteq(S, \rho) .\) Prove that \(B\) is connected in \(S\) iff it is connected in \((A, \rho) .\)
Suppose that no two of the sets \(A_{i}(i \in I)\) are disjoint. Prove that if all \(A_{i}\) are connected, so is \(A=\bigcup_{i \in I} A_{i}\).
[Hint: If not, let \(A=P \cup Q(P, Q \text { as in Definition } 3) .\) Let \(P_{i}=A_{i} \cap P\) and \(Q_{i}=A_{i} \cap Q,\) so \(A_{i}=P_{i} \cup Q_{i}, i \in I .\)
At least one of the \(P_{i}, Q_{i}\) must be \(\emptyset \text { (why?); say, } Q_{j}=\emptyset \text { for some } j \in I . \text { Then }\)
\((\forall i) Q_{i}=\emptyset,\) for \(Q_{i} \neq \emptyset\) implies \(P_{i}=\emptyset,\) whence
\[
A_{i}=Q_{i} \subseteq Q \Longrightarrow A_{i} \cap A_{j}=\emptyset\left(\text { since } A_{j} \subseteq P\right) ,
\]
\(\left.\text { contrary to our assumption. Deduce that } Q=\bigcup_{i} Q_{i}=\emptyset . \text { (Contradiction!) }\right]\)
Prove that if \(\left\{A_{n}\right\}\) is a finite or infinite sequence of connected sets and if
\[
(\forall n) \quad A_{n} \cap A_{n+1} \neq \emptyset ,
\]
then
\[
A=\bigcup_{n} A_{n}
\]
is connected.
[Hint: Let \(B_{n}=\bigcup_{k=1}^{n} A_{k} .\) Use Problem 9 and induction to show that the \(B_{n}\) are connected and no two are disjoint. Verify that \(A= \bigcup_{n} B_{n}\) and apply Problem 9 to \(\left.\text { the sets } B_{n} .\right]\)
Given \(p \in A, A \subseteq(S, \rho),\) let \(A_{p}\) denote the union of all connected subsets \(\text { of } A \text { that contain } p \text { (one of them is }\{p\}) ; A_{p}\) is called the \(p\)-component of \(A .\) Prove that
\(\left.\text { (i) } A_{p} \text { is connected (use Problem } 9\right)\);
(ii) \(A_{p}\) is not contained in any other connected set \(B \subseteq A\) with \(p \in B\);
(iii) \((\forall p, q \in A) A_{p} \cap A_{q}=\emptyset\) iff \(A_{p} \neq A_{q} ;\) and
(iv) \(A=\cup\left\{A_{p} | p \in A\right\}\).
[Hint for (iii): If \(A_{p} \cap A_{q} \neq \emptyset\) and \(A_{p} \neq A_{q},\) then \(B=A_{p} \cup A_{q}\) is a connected set \(\left.\text { larger than } A_{p}, \text { contrary to (ii). }\right]\)
Prove that if \(A\) is connected, so is its closure (Chapter 3, §16 Definition 1 ), and so is any set \(D\) such that \(A \subseteq D \subseteq \overline{A}\).
[Hints: First show that \(D\) is the "least" closed set in \((D, \rho)\) that contains \(A\) \(\text { (Problem } 11 \text { in Chapter } 3, §16 \text { and Theorem } 4 \text { of Chapter } 3, §12) .\) Next, seeking a contradiction, let \(D=P \cup Q, P \cap Q=\emptyset, P, Q \neq \emptyset,\) clopen in \(D .\) Then
\[
A=(A \cap P) \cup(A \cap Q)
\]
proves \(A\) disconnected, for if \(A \cap P=\emptyset,\) say, then \(A \subseteq Q \subset D\) (why?), contrary to \(\text { the minimality of } D ; \text { similarly for } A \cap Q=\emptyset .]\)
A set is said to be totally disconnected iff its only connected subsets are one-point sets and \(\emptyset\).
Show that \(R\) (the rationals) has this property in \(E^{1}\).
Show that any discrete space is totally disconnected (see Problem 13).
From Problems 11 and 12 deduce that each component \(A_{p}\) is closed \(\left(A_{p}=\overline{A_{p}}\right) .\)
Prove that a set \(A \subseteq(S, \rho)\) is disconnected iff \(A=P \cup Q,\) with \(P, Q \neq \emptyset,\) and each of \(P, Q\) disjoint from the closure of the other: \(P \cap \overline{Q}=\emptyset=\overline{P} \cap Q .\)
[Hint: By Problem 12, the closure of \(P\) in \((A, \rho)\) (i.e., the least closed set in \((A, \rho)\) that contains \(P\) ) is
\[
A \cap \overline{P}=(P \cup Q) \cap \overline{P}=(P \cap \overline{P}) \cup(Q \cap \overline{P})=P \cup \emptyset=P ,
\]
so \(P\) is closed in \(A ;\) similarly for \(Q .\) Prove the converse in the same manner.
Give an example of a connected set that is not arcwise connected.
[Hint: The set \(G_{0 b}(a=0)\) in Problem 4 is the closure of \(G_{0 b}-\{\overline{0}\}\) (verify!), and \(\left.\text { the latter is connected (why?); hence so is } G_{0 b} \text { by Problem } 12 .\right]\)