c) given an arbitrary convex function wcx) :R"→tR, prove fhat is a convex set.

Q: Is R, equipped with the finite-closed topology, Hausdorff? Provide a proof supporting your answer.

A: Given: ℝ is equipped with the finite-closed topology. Let, ℝ with this topology be Hausdorff and x,…

Q: A(n)................................... is the collection of all points in the plane such that the…

A: We know, A parabola is a collection of all the points in a plane such that the distance from each…

Q: Let A be a set and let ƒ : A → B be a surjective function. Prove that there exists a subset CCA such…

A: Given: A be a set and f:A→B is a surjective function. To prove: There exists a subset C⊂A such that…

Q: Consider the set P = {f€C([0, 1], R) | ƒ is nondecreasing} Determine if (P,d2) is separable. Here,…

A:

Q: Suppose A ⊂ R is both closed and bounded, prove that A has the Heine-Borel Property

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Q: Suppose that l is a line. Let H be one of the two half-planes bounded by l. Give a proof that l U H…

A: Answer is mentioned below

Q: Let X be any set with the discrete topology. Show that X is normal.

A:

Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…

Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: Let A be any set with the discrete topology. Show that A is normal.

A:

Q: In a normed linear space X, prove that the closed ball, ū (x, r) is a convex set for any хE X, and…

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Q: of all Z € C that z iz e = e a Descnibe and sketch the set

A:

Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: If M is a closed linear subs r space X, and x, be a vect the distance from x, to M, ional f, in X'…

A:

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

A:

Q: Let C and D be finite sets and let f: C D and g: D C be two functions such that Vy eD f(g(y)) = y…

A: If there exists an one-one map from D to C then "|D| < |C|" or "|C| >|D|".

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: Find an example to show that the convexity of S is necessary in Exercise 19. Reference Exercise 19:…

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: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: Let C and D be finite sets and let f :C → D and g : D →C be two functions such that Vy ED f(g(y)) =…

A:

Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

A:

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let A be a nonempty set and define B={kx : x∈A}, where k≥0 is fixed. Prove that supB=ksupA.

A: Given A be a nonempty set and define B=kx:x∈A where k≥0 is fixed. We have to prove that supB=ksupA…

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

A:

Q: a. Show that the null-space N(A) is a convex set. Is it true for every A?

A: As both questions are of different type,we shall solve first Que only. For other questions kindly…

Q: Let V C(R ) be a nonempty open set satisfying a – b EV whenever a, b e V. Prove that V = R. %D

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Q: Find the transitive closure of R if M. is

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Q: Let X be an uncountable set and let T = {U C X :U = Ø or U¢ is finite} %3D Then is topological space…

A: let X be an uncountable set and T=U⊑X : U=∅ or Uc is finite i.e cofinite topology on X. Then the…

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x>2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

A:

Q: Prove by sequences that the open balls in R^n with the usual (Euclidean) metric are not compact

A: In this question, we have to show that open balls in ℝn with the usual metric are not compact by the…

Q: Use the Heine-Borel theorem to show that every closed and bounded interval [a, b] with the absolute…

A: Compact Set: A collection A of subsets of space X is said to be cover X or covering of X if the…

Q: Let F and G be nonempty bounded sets. Prove that inf (F UG) = min{inf F, inf G} and sup (F UG) =…

A: Since F and G are non-empty bounded subsets of R, sup F, sup G, inf F and inf G are all exist in R.…

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

A:

Q: Sketch the following sets of points in the x-y plane. {(x, x^2): x is in R}

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Q: If S1 and S2 are convex sets, prove that their intersection S1 N S2 is also a convex set.

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Q: a nonempty compact set of IR. sup Ke K. be

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Q: ndary of a path-connected set in R^2 with the usual

A: We have to check that boundary of a path connected set in R2 with the usual topology is…

Q: Is the set θ={((x,y),(3y,2x,x+y)) : x,y∈R} a function? If so, what is its domain and range? What can…

A:

Q: Let x1 and x2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: (b) (i) Give an example of an infinite collection of nested open sets in R under the usual metric on…

A: Here given usual metric on the set of real numbers R

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.