Topology

Abstract

This chapter contains a brief introduction to point set topology. The main aim is to extend some of the important results about continuous functions on a compact intervals to continuous functions on a larger class of sets, the compact sets.

Appendices

Problems

1.1 Problems for Sect. 13.1

Infinite intersections of open sets need not be open sets:

1. Suppose $\mathbb {K}=\mathbb {R}.$ Prove that $\bigcap _{n=1}^{\infty }\left ]0,1+\frac {1}{n}\right [$ is an intersection of open sets that is not an open set.

2. Construct a sequence $A_{n}$ of open sets in $\mathbb {R}^{2},$ such that $\bigcap _{n=1}^{\infty }A_{n}$ is not an open subset of $\mathbb {R}^{2}.$

3. The triangle $\{(x,y)\in \mathbb {R}^{2}\mid x>0,y>0,x+y<1\}$ is an open set in $\mathbb {R}^{2}$.

4. Show that $\bigcap _{n=1}^{\infty }B_{1+\frac {1}{n}}(x)=\overline {B}_{1}(x).$

5. The set $\mathbb {R}\setminus \left \{ \frac {1}{n}\mid n\in \mathbb {N}\right \}$ is not an open subset of $\mathbb {R},$ because any interval of the form $B_{r}(0)=]-r,r[$ will contain all $\frac {1}{n}$ with $n>\frac {1}{r}.$

6. Show the set $\mathbb {R}\setminus \left (\left \{ \frac {1}{n}\mid n\in \mathbb {N}\right \} \cup \{0\}\right )$ is an open subset of $\mathbb {R}.$

7. If $C$ is the Cantor set, then $[0,1]\setminus C$ is an open subset of $\mathbb {R}.$

8. If $C$ is the Cantor set, then $\overset {\circ }{C}=\emptyset .$

9. If $A$ and $B$ are open subsets of the real line $\mathbb {R},$ then $A\times B$ is an open subset of the plane $\mathbb {R}^{2}.$

1.2 Problems for Sect. 13.2

1. If $A$ and $B$ are closed subsets of the real line $\mathbb {R},$ then $A\times B$ is an closed subset of the plane $\mathbb {R}^{2}.$

2. The Cantor set is a closed subset of $\mathbb {R}.$

3. Find a sequence of closed sets $K_{n},$ such that $\bigcup _{n=1}^{\infty }K_{n}$ is not a closed set.

4. Find a sequence of closed sets $K_{n},$ such that $\bigcup _{n=1}^{\infty }K_{n}$ is an open set.

5. Establish the following:

(i) $\overline {Q}=\mathbb {R}$

(ii) $\overline {\mathbb {Q}\times \mathbb {Q}}=\mathbb {R}\times \mathbb {R}.$

6. The closure of the open ball $B_{r}(x)$ is the closed ball $\overline {B}_{r}(x).$

7. If $C$ is the Cantor set, then $\overline {\mathbb {R}\setminus C}=\mathbb {R}.$

8. For any sets $A,B$ we have $\overline {A\cup B}=\overline {A}\cup \overline {B}.$

9. For any sets $A,B$ we have $\overline {A\cap B}\subseteq \overline {A}\cap \overline {B}.$

10. If $A:=\mathbb {Q}$ and $B:=\mathbb {R}\setminus \mathbb {Q},$ then $\overline {A\cap B}=\emptyset$ and $\overline {A}\cap \overline {B}=\mathbb {R}.$

The boundary of the set $D$ is the set $\partial D:=\overline {D}\setminus \overset {\circ }{D}.$

11. $\partial [a,b]=\{a,b\}.$

12. $\partial B_{r}(a)=\{x\mid |x-a|=r\}.$

13. If $C$ is the Cantor set, then $\partial C=C.$

14. If $C$ is the Cantor set, then $\partial \left (\mathbb {R}\setminus C\right )=C.$

1.3 Problems for Sect. 13.3

1. Find an open cover of $\left [0,1\right ]\cap \mathbb {Q}$ that does not have a finite subcover.

2. Give an example of an increasing sequence $\left (a_{n}\right )$ such that the set $\left \{ a_{n}\mid n\in \mathbb {N}\right \}$ is compact.

3. If $a_{n}<a_{n+1}$ for all $n,$ then the set $\left \{ a_{n}\mid n\in \mathbb {N}\right \}$ is not compact.

4. Let $\left (z_{n}\right )$ be a convergent sequence with limit $\widetilde {z}.$ The set $\left \{ z_{n}\mid n\in \mathbb {N}\right \}$ is compact iff $z_{n}=\widetilde {z}$ for some $n.$

5. Use the definition of covering compactness to show that if $a$ and $B$ are covering compact, then the union of $a$ and $B$ is also covering[AQ2] compact.

1.4 Problems for Sect. 13.4

1. Suppose $f:A\to B$ is continuous. If $D$ is a pathwise connected subset of $A,$ then $f(D)$ is a pathwise connected subset of $B.$

2. Suppose $f:A\to \mathbb {R}$ is continuous and $a$ is pathwise connected. Let $a,b$ be points in $A,$ such that $f\left (a\right )<f\left (b\right ).$ If $f\left (a\right )<k<f\left (b\right ),$ show there is a $C$ in $A,$ such that $f\left (c\right )=k.$

3. Let $I$ be an interval and suppose $f:I\to \mathbb {R}$ is differentiable. Let

$$ D:=\left\{ f'(x)\mid x\in I\right\} $$

and

$$ C:=\left\{ \frac{f(b)-f(a)}{b-a}\mid a,b\in I,a<b\right\} . $$

(a) Prove $C\subseteq D\subseteq \overline {C}.$

(b) Prove $C$ is an interval. [Hint: One way is to prove $C$ is pathwise connected. To this end, note for $x,y$ in $I,$ the function $\varphi _{x,y}(t):=\left (1-t\right )x+ty$ determines a path in $I$ beginning at $\varphi _{x,y}(0)=x$ and ending at $\varphi _{x,y}(1)=y.$ Show

$$ \psi(t):=\frac{f\left(\varphi_{b,d}(t)\right)-f\left(\varphi_{a,c}(t)\right)}{\varphi_{b,d}(t)-\varphi_{a,c}(t)},0\leq t\leq1 $$

is a path in $C$ connecting $\frac {f(b)-f(a)}{b-a}$ to $\frac {f(d)-f(c)}{d-c}$ for all $a<b$ and all $c<d$ in $I.$]

Solutions and Hints for the Exercises

Exercise 13.1.4. By Example 13.1.1.$B_{r}(x)$ is in the interior of the closed ball $\overline {B_{r}}(x).$ If $|x-y|=r$ and $s>0,$ then $B_{s}(y)$ is not a subset of $B_{r}(x).$

Exercise 13.1.5. Let $x\in \overset {\circ }{A}.$ Pick $r>0$ such that $B_{r}(x)\subseteq A.$ Let $y\in B_{r}(x).$ For some $s>0,$ $B_{s}(y)\subseteq B_{r}(x).$ Hence, $y\in \overset {\circ }{A}.$

Exercise 13.2.5 If $a$ is closed in $D,$ then $A=D\cap K$ for some closed set $K.$ Hence, $D\setminus A=D\setminus (D\cap K)=D\cap (\mathbb {C}\setminus K).$ The converse is similar.

Exercise 13.2.6. By Exercise 13.2.5$f^{-1}(K)$ is closed in $D$ iff $\mathbb {C}\setminus f^{-1}(K)=f^{-1}(\mathbb {C}\setminus K)$ is open in $D.$

Exercise 13.2.7. If $|x-y|=r$ then $B_{s}(y)\cap B_{r}(x)\neq \emptyset .$ Hence, $y$ is an accumulation point of $B_{r}(x).$

Exercise 13.2.8. If $K$ is closed, then $K'\subseteq K.$ Hence $K\cup K'=K.$

Exercise 13.2.9. Let $\widehat {K}$ be the set of contact points of $K.$ Any point in $K$ is a contact point of $K$ and any accumulation point of $K$ is a contact point of $K,$ hence $\overline {K}\subseteq \widehat {K}.$ Conversely, if $x\notin K$ is a contact point of $K,$ then $x$ is an accumulation point of $K.$ Consequently, $\widehat {K}\subseteq \overline {K}.$

Exercise 13.3.4. If $K$ is not bounded, then $\left (B_{1}(k)\right )_{k\in K}$ is an open cover without a finite subcover.

Exercise 13.3.6. If $K$ is not closed, then $K$ has an accumulation point $a$ that is not in $K.$ Let $A_{0}$ be the complement of the closed ball $\overline {B_{1/2}}\left (a\right )$ and let $A_{n}=B_{1/n}(a)$ for $n\in \mathbb {N}.$ Then $\left (A_{n}\right )_{n\in \mathbb {N}_{0}}$ is an open cover of $K$ without a finite subcover.

Exercise 13.3.16. Let $x_{1}\in K,$ $x_{2}\in K$ with $|x_{2}|>1+|x_{1}|,$ $x_{3}\in K$ with $|x_{3}|>1+|x_{2}|,$ and so on. Then $|x_{k}|\to \infty .$ Hence $(x_{k})$ is not convergent.

Exercise 13.3.18. Suppose $K$ is sequentially compact and $a$ is an accumulation point of $K.$ For each $n,$ let $x_{n}\in B_{1/n}(a)\cap \left (K\setminus \{a\}\right ).$ Then $x_{n}\to a.$ Hence, any subsequence of $(x_{n})$ converges to $a.$ Thus $a\in K.$

Exercise 13.3.19. By definition of convergence, $x_{0}$ is a contact point of A.

Exercise 13.4.1. $\implies$ is a consequence of the Intermediate Value Theorem and the Interval Theorem. $\Longleftarrow$ Suppose $D$ is an interval and $a<b$ are in $D.$ Then $f(t):=a+t(b-a)$ is the required path.

Exercise 13.4.2. If $(a,b)$ and $(\alpha ,\beta )$ are in $[0,1]^{2},$ then $f(t):=(a,b)+t(\alpha -a,\beta -b)$ is the required path.