Chapter 4

Exercise \(4.3 .1 .\) Identify the limit points and isolated points of the following sets: a. [-1,1] , b. (-1,1) c. \(\left\\{\frac{1}{n}: n \in \mathbb{Z}^{+}\right\\}\), d. \(\mathbb{Z}\) e. \(\mathbb{Q}\).

Show that if \(A \subset \mathbb{R},\) then \(A^{\circ}\) is open.

Show that every finite subset of \(\mathbb{R}\) is compact.

Suppose \(x\) is a limit point of the set \(A\). Show that for every \(\epsilon>0,\) the set \((x-\epsilon, x+\epsilon) \cap A\) is infinite.

Suppose \(n \in \mathbb{Z}^{+}\) and \(K_{1}, K_{2}, \ldots, K_{n}\) are compact sets. Show that \(\bigcup_{i=1}^{n} K_{i}\) is compact.

Show that every closed interval \(I\) is a closed set.

Let \(U \subset \mathbb{R}\) be a nonempty open set. Show that \(\sup U \notin U\) and \(\inf U \notin U\).

Show that if \(K\) is compact and \(C \subset K\) is closed, then \(C\) is compact.

For \(n=1,2,3, \ldots,\) let \(I_{n}=\left(-\frac{1}{n}, \frac{n+1}{n}\right) .\) Is $$ \bigcap_{n=1}^{\infty} I_{n} $$ open or closed?

For \(n=3,4,5, \ldots,\) let \(I_{n}=\left[\frac{1}{n}, \frac{n-1}{n}\right] .\) Is $$ \bigcup_{n=3}^{\infty} I_{n} $$ open or closed?

Show that a set \(K \subset \mathbb{R}\) is compact if and only if every infinite subset of \(K\) has a limit point in \(K\).

Suppose, for \(n=1,2,3, \ldots,\) the intervals \(I_{n}=\left[a_{n}, b_{n}\right]\) are such that \(I_{n+1} \subset I_{n} .\) If \(a=\sup \left\\{a_{n}: n \in \mathbb{Z}^{+}\right\\}\) and \(b=\inf \left\\{b_{n}: n \in \mathbb{Z}^{+}\right\\},\) show that $$ \bigcap_{n=1}^{\infty} I_{n}=[a, b] $$

Find a sequence \(I_{n}, n=1,2,3, \ldots,\) of closed intervals such that \(I_{n+1} \subset I_{n}\) for \(n=1,2,3, \ldots\) and $$ \bigcap_{n=1}^{\infty} I_{n}=\emptyset $$

Suppose \(K_{1}, K_{2}, K_{3}, \ldots\) are nonempty compact sets with $$ K_{n+1} \subset K_{n} $$ for \(n=1,2,3, \ldots .\) Show that $$ \bigcap_{n=1}^{\infty} K_{n} $$ is nonempty.

Find a sequence \(I_{n}, n=1,2,3, \ldots,\) of bounded, open intervals such that \(I_{n+1} \subset I_{n}\) for \(n=1,2,3, \ldots\) and $$ \bigcap_{n=1}^{\infty} I_{n}=\emptyset $$

We say a collection of sets \(\left\\{D_{\alpha}: \alpha \in A\right\\}\) has the finite intersection property if for every finite set \(B \subset A\), $$ \bigcap_{\alpha \in B} D_{\alpha} \neq \emptyset $$ Show that a set \(K \subset \mathbb{R}\) is compact if and only for any collection \(\left\\{E_{\alpha}: \alpha \in A, E_{\alpha}=C_{\alpha} \cap K\right.\) where \(C_{\alpha} \subset \mathbb{R}\) is closed \(\\}\) which has the finite intersection property we have $$ \bigcap_{\alpha \in A} E_{\alpha} \neq \emptyset $$

Suppose \(A_{i} \subset \mathbb{R}, i=1,2, \ldots, n,\) and let \(B=\bigcup_{i=1}^{n} A_{i} .\) Show that $$ \bar{B}=\bigcup_{i=1}^{n} \overline{A_{i}} $$

Suppose \(A_{i} \subset \mathbb{R}, i \in \mathbb{Z}^{+},\) and let $$ B=\bigcup_{i=1}^{\infty} A_{i} $$ Show that $$ \bigcup_{i=1}^{\infty} \overline{A_{i}} \subset \bar{B} $$ Find an example for which $$ \bar{B} \neq \bigcup_{i=1}^{\infty} \overline{A_{i}} $$

Suppose \(U \subset \mathbb{R}\) is a nonempty open set. For each \(x \in U,\) let $$ J_{x}=\bigcup(x-\epsilon, x+\delta) $$ where the union is taken over all \(\epsilon>0\) and \(\delta>0\) such that \((x-\epsilon, x+\delta) \subset U\). a. Show that for every \(x, y \in U,\) either \(J_{x} \cap J_{y}=\emptyset\) or \(J_{x}=J_{y}\). b. Show that $$ U=\bigcup_{x \in B} J_{x} $$ where \(B \subset U\) is either finite or countable.