6. Give an example of a set S which is (i) bounded above but not bounded below, (ii) bounded below but not bounded above, (iii) neither bounded above nor bounded below,

Q: Set A is bounded above if and only if set -A is bounded below.

A:

Q: 4. Give an example of a non-measurable subset of R. (witheutProof)

A:

Q: Prove or disprove. A closed and bounded subset of e' is compact.

A: PROVE OR DISPROVE :-  A closed and bounded subset of l1 is compact.

Q: 3.1 Show that a finite intersection of closed sets is closed.

A:

Q: that the boundary of a closed set is nowhere dense in

A: Matric space

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

A:

Q: 26

A: The objective is to find an infinite collection of closed sets whose union is not closed.

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

A:

Q: 5. Give an example (by words) for a collection of objects which is not well defined

A: Given, We have to write an example for a collection of objects which is not well defined.

Q: S is a nonempty set of real numbers that is bounded.

A:

Q: Let A be a bounded non-empty set. Then inf A < sup A. True False

A:

Q: 5. Show that if A and B are bounded subsets of R, then A U B is a bounded set. Show that sup(A U B)…

A:

Q: 2. Show that any subset of a countable set is countable. 3. Show that a nonempty set S is countable…

A: Hence any subset of a countable set is countable.

Q: Prove that if S is a bounded set then S is bounded.

A:

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

A:

Q: 4) The relationk onthe Selof real numbers Risa partialorder relation.

A: relation R on a set A is called a partial order relation if it satisfies the following three…

Q: A set SCR" is bounded if there exists k €R such that, for all x E S, || = VE, 0. What can you say…

A:

Q: 4. Let A be an open set in å nörmed space subset B of X .

A: Que  Ans :  Given ( X, ||-||) be a norm linear space,  A be any open subset of X    ( def- set A is…

Q: 8. (a) Show that the intersection of finitely many open sets is open.

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: rcise: Prove that every finite set S of R" has no accumulation point.

A: Proof of the the given theorem is given as

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: 3.6 Prove that a closed set in the metric space (S, d) either is nowhere dense in S or else contains…

A:

Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: Prove that the unoin of a finite number of bounded sets is bounded.

A:

Q: 5.3.8 Prove that if A CR is a finite set, then A is well-ordered.

A:

Q: Prove that derived set of a bounded set is also bounded.

A:

Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: Prove that a nonempty set is bounded if and only if it is contained in a bounded interval

A: A non empty set S is called bounded if it has both upper and lower bounds, In a certain sense a non…

Q: let to be anonempty subset of abounded set E-Show that infESinf to Ssup E ssup E

A: Given :  E0 is a non empty subset of a bounded set E. To show :  inf E ≤ inf E0 ≤sup E0 ≤ sup E

Q: Prove that every nonempty subset of a well-ordered set is well-ordered.

A: Given problem is :

Q: Show that any subset of bounded set is bounded

A:

Q: A fuzzy set B is closed if and only if B contains all the accumulation point of B. accumulation…

A: Option-A is correct.

Q: Prove the following. 1. Bolzano-Weierstrass Theorem: If a bounded set S in R" contains infinitely…

A:

Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: (a) The intersection of an arbitrary collection of closed sets is closed. (b) The union of any…

A: (a) We have to prove that the intersection of an arbitrary collection of closed sets is closed. Let…

Q: 5. If X any Y are sets, then X-Y {r€X: z4Y} If X is open set and Y is closed set. Prove that (a) X-Y…

A: Let X is open set and Y is closed set

Q: 7. Suppose that a set X is equipped with the finite closed topology. Which one of the followings is…

A: As per the policy, we are solving first question. Please repost the question and specify which…

Q: 5. (a) How many functions are there from a set with n elements to a set with m elements?

A:

Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: )Š is the smallest closed set in M such that S CS.

A:

Q: ave a class boundary that is less than zero?  On the probelem I am working, the class goes from 0-5

A: Class boundaries are the values which help in separating the data points in statistics.

Q: 3. If is a collection of dosed sets in R such that every finite subcollection has a nonempty…

A: Given : θ is a collection of closed sets in ℝ such that every finite subcollection has a non empty…

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

A:

Q: 2. Prove that the union of any finite collection of closed sets in RP is a closed set in RP.

A: A set A is closed if and only if every limit point of A belongs to A. Now we can prove the result.…

Q: 2- Moke the definitions of open and closed sets. x =Larbic,d,eļ and ノ closed subsets of (x,T) space,…

A:

Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

A:

Q: prove that every subset of negligible. set is negligible.

A: I have used definition of negligible set.