Problem Set based on VI. Convergence, open and closed sets.

Problem 1.

Give an example of a metric space R and two open spheres $S(x,r_{1})$ , $S(y, r_{2})$ in R such that $S(x, r_{1}) \subset S(y,r_{2})$ although $r_{1}> r_{2}$ .

Problem 2:

Prove that every contact point of a set M is either a limit point of M or is an isolated point of M.

Comment. In particular, $[M]$ can only contain points of the following three types:

a) Limit points of M belonging to M.

b) Limit points of M which do not belong to M.

c) Isolated points of M.

Thus, $[M]$ is the union of M and the set of all its limit points.

Problem 3:

Prove that if $x_{n}\rightarrow x$ and $y_{n} \rightarrow y$ as $n \rightarrow \infty$ then $\rho(x_{n},y_{n}) \rightarrow \rho(x,y)$ .

Hint : use the following problem: Given a metric space $(X,\rho)$ prove that $|\rho(x,z)-\rho(y,u)| \leq |\rho(x,y)|+|\rho(z,u)|$

Problem 4:

Let f be a mapping of one metric space X into another metric space Y. Prove that f is continuous at a point $x_{0}$ if and only if the sequence $\{ y_{n}\} = \{ f(x_{n})\}$ converges to $y=f(x_{0})$ whenever the sequence $x_{n}$ converges to $x_{0}$ .

Problem 5:

Prove that :

(a) the closure of any set M is a closed set.

(b) $[M]$ is the smallest closed set containing M.

Problem 6:

Is the union of infinitely many closed sets necessarily closed? How about the intersection of infinitely many open sets? Give examples.

Problem 7:

Prove directly that the point $\frac{1}{4}$ belongs to the Cantor set F, although it is not the end point of any of the open interval deleted in constructing F. Hint: The point $\frac{1}{4}$ divides the interval $[0,1]$ in the ratio $1:3$ and so on.

Problem 8:

Let F be the Cantor set. Prove that

(a) the points of the first kind form an everywhere dense subset of F.

(b) the numbers of the form $t_{1}+t_{2}$ where $t_{1}, t_{2} \in F$ fill the whole interval $[0,2]$ .

Problem 9:

Given a metric space R, let A be a subset of R, and $x \in R$ . Then, the number $\rho(A,x) = \inf_{a \in A}\rho(a,x)$ is called the distance between A and x. Prove that

(a) $x \in A$ implies $\rho(A,x)=0$ but not conversely

(b) $\rho(A,x)$ is a continuous function of x (for fixed A).

(d) $[A]=A \bigcup M$ , where M is the set of all points x such that $\rho(A,x)=0$ .

Problem 10:

Let A and B be two subsets of a metric space R. Then, the number $\rho(A,B)= \inf_{a \in A, b \in B}\rho(a,b)$ is called the distance between A and B. Show that $\rho(A,B)=0$ if $A \bigcap B \neq \phi$ , but not conversely.

Problem 11:

Let $M_{K}$ be the set of all functions f in $C_{[a,b]}$ satisfying a Lipschitz condition, that is, the set of all f such that $|f(t_{1}-f(t_{2})| \leq K|t_{1}-t_{2}|$ for all $t_{1}, t_{2} \in [a,b]$ , where K is a fixed positive number. Prove that:

a) $M_{K}$ is closed and in fact is the closure of the set of all differentiable functions on $[a,b]$ such that $|f^{'}(t)| \leq K$

(b) the set $M = \bigcup_{K}M_{K}$ of all functions satisfying a Lipschitz condition for some K is not closed;

Problem 12:

An open set G in n-dimensional Euclidean space $R^{n}$ is said to be connected if any points $x,y \in G$ can be joined by a polygonal line(by a polygonal line we mean a curve obtained by joining a finite number of straight line segments end to end.) lying entirely in G. For example, the open disk $x^{2}+y^{2}<1$ is connected, but not the union of the two disks $x^{2}+y^{2}<1$ , $(x-2)^{2}+y^{2}<1$ (even though they share a contact point). An open subset of an open set G is called a component of G if it is connected and is not contained in a larger connected subset of G. Use Zorn’s lemma to prove that every open set G in $R^{n}$ is the union of no more than countably many pairwise disjoint components.

Comment: In the case $n=1$ , that is, the case on the real line, every connected open set is an open interval, possibility one of the infinite intervals $(-\infty, \infty)$ , $(a, \infty)$ or $(-\infty, b)$ . Thus, theorem 6 (namely: Every open set G on the real line is the union of a finite or countable system of pairwise disjoint open intervals) on the structures of open sets on the line is tantamount to two assertions:

(i) Every open set on the line is the union of a finite or countable number of components.

(ii) Every open connected set on the line is an open interval.

The first assertion holds for open sets in $R^{n}$ (and, in fact, is susceptible to further generalizations), while the second assertion pertains specifically to the real line.

Cheers,

Happy analysis !!

Nalin Pithwa

MScMathematics

Problem Set based on VI. Convergence, open and closed sets.

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