A set L cR is convex if:

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

A:

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

Q: Let A and B be subsets of a set X. Prove that if A ⊆ Bc, then A and B are disjoint.

A:

Q: 6. Show that the span of this set is all of R?. {}}

A: Let S=v1,v2,...,vn be a set of vectors in a vector space V(F) . The vector space V is spanned by S,…

Q: A metric space (X , d) and a collection of open sets U such that UEU is not open.

A: Let X, d be a metric space. Let U be the collection of open sets in the metric space X, d. That is,…

Q: If S is compact and D is in S is a closed set, then D is compact. Use the definition of…

A: Let S is compact and D be a closed subset of S. We are to prove that the set D is compact in S. Now…

Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

A:

Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: Prove that intersection of any class of convex subsets of N is convex?

A:

Q: Determine whether is a linearly independent set of vec- 3 11 tors in R3.

A: A set of vectors v1,v2,v3,…,vk in a vector space is said to be linearly independent if…

Q: Find an example of nonempty, distinct sets A, B, C for which (ANB) UC = (AUB) nC.

A:

Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

A:

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

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: Choose any that is a subset of the set (Ø} {{Ø),0} {{1}}

A:

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

A:

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

A:

Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

A:

Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

A: Let A be a non-empty compact subset of ℝ Let B be a non-empty closed subset of ℝ such that A∩B=∅ To…

Q: Given sets A and B, how many constant maps are there from A into B?

A: Given Set A and B Let |A| = m |B| =n

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: Given sets A and B, how many constant maps are there from A into B? Give an example.

A:

Q: Is the Cantor set closed? Is it compact? Explain!

A:

Q: The set X is an open set in the ambient metric space . O True O False

A: Solution

Q: - If C is Convex set ond a GR, then XC is a Convex set

A: Let C be a convex set and x∈R Then we have to show that xC is a convex set. The set xC is defined as…

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

A: Given the usual topology on ℝ3. i.e. τusual=U⊆ℝ3: ∀x∈U, ∃∈>0 such that B∈x⊆U

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: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper…

A: Let X be a non empty set. Let a∈X. Then a is said to be a lower bound of the set X if a≤x for all…

Q: Is there a set A that satisfies A = {A}? If yes, exhibit one such. If not, Why not ?

A:

Q: If A and B are connected subsets of RP, give examples to show that AU B, An B, A\B can be either…

A: If A and B are connected subsets of ℝp,give example to showthat A∪B, A∩B, A/B can be either…

Q: How many subsets are there for the set {h, j}?

A: The given set is {h, j}.It is known that, “If a set contains ‘m’ elements, then the number of…

Q: Let X be a set. Show that P(X)÷ X.

A: Solution:Let X ba a set.To Prove: P(X)≠X

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

A:

Q: A set of ordered pairs in which each member of the set of first components corresponds to exactly…

A:

Q: Suppose A={-1, 0, 4}.Find a set B such that A is equivalent to B.

A: Two sets A and B are said to be equivalent to each other if both the sets A and B have same number…

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: Show that the derived set E′ of any set E is closed.

A: Here is the proof .

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: For any metric space (X, d) and x EX, the singleton set (x) is closed. Select one: OTrue O False

A:

Q: Suppose {p, p,, P3} is an affinely independent set in Rn and q is an arbitrary point in Rn. Show…

A: Here, {p1,p2,p3} is an affinely independent set. So, {p1,p2,p3} are linearly independent.

Q: Let A and B be two finite sets such that |A| = 20, |B| = 28 and |A U BI

A: Given A=20B=28A∪B=36

Q: Show the following set is convex. x≤10 [(1+(x-10)-²)¹ x>10 Determine a-cut sets of the above set for…

A: To show that the given set is convex: μAx=0                               x≤101+x-10-2-1  x>10…

Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

A:

Q: Determine if the set S is span set of %3D R2x2. Solution:

A:

Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: Give example for following 1/ set is not compact

A: Compact set:- A set S⊆R is called compact “if every sequence in S has a subsequence that converges…

Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…