# 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).

(c) $\rho(A,x)=0$ if and only if $x$ is a contact point of 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;

(c) The closure of M is the whole space $C_{[a,b]}$

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

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