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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