Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

Q: 2. Let (S, d) be a discrete metric space and p E S. Identify S i(P), S1 (p), S2(p).

A: We shall provide answers for first question only as per the answering guidelines. For other…

Q: If τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normal space…

A: τ1 = {A ⊂ X : p ∈ A} ∪ {∅} is a topology on X, then determine whether (X, τ1) is a normalspace or…

Q: Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects…

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Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

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Q: Let A = {0, 1} with the discrete topology and let f: A R be defined by f (0) = -1, f(1) = 1.

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Q: Let (X, F) be a o – algebra | , and be two probability measures on (X, F) so that both are ergodic…

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Q: The only subsets of R which are both open and closed under the discrete metric are Ø and R. Select…

A: This is False statement, in discreate metric space, every subset of R is both open as well as…

Q: be a closed subset of space Xa

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Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

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

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Q: Let E be a subset of X, where X is a metric space. Show the limit points of E are the same as the…

A: In mathematics, the closure of a subset S of points in a topological space consists of all points in…

Q: Let (X,T) be a discrete space, then Va e X, the set {a} is а. орen b. closed c. clopen d. is neither…

A: Given problem is :

Q: Let (X, E) be a measurable space and (En) be a sequence of pairwise disjoint sets in E. If, for each…

A: Given X,∑ is a measurable space and En  be a sequence of pairwise disjoint sets in ∑. Given f is…

Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

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Q: Let A be a subset of a topological space X. Show that bd(A) = Ø iff A is open and closed in X.

A: We need to prove the statement.

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

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

A: Solution

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: if (X,d) is discrete metric space, ACX and A+D (x,d) Find the sets of A° and A (functional analysis…

A: Consider a discrete metric space X,d where d is defined as: dp,q=0,p=q1,p≠q

Q: Let (2, F,µ) be a measure space. (a). Let A, B E F such that A C B. Prove that u(A) sH(B). (b) Let…

A: Given that Ω,F,μ be a measure space. We know that Ω,F,μ is a measure space if: μϕ=0…

Q: Prove the following. Let J be any set, and let {Xa : a E J} be an indexed family of connected…

A: It is given that Xα:α∈J is an indexed family of connected topological space that is each component…

Q: Let {A :A E A} be a family of subsets of IR. Show that • When A is finite, we have

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Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

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Q: Exercise 6. Let ECR such that E is nonempty and bounded. Prove that the set -E= {-x : x E E}…

A: NOTE:Kindly repost the other questions as a separate question.

Q: Find the image of the given set under the exponential mapping The semi-infinite strip ∞o < x≤0,0 ≤ y…

A: Given That : Exponential mapping  the semi-infinite stripe ∞&lt;x≤0, 0≤y≤π To Find :Image the given…

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

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Q: Let (X, A.u) be a measure space defined as µ(A) = { 1 if a e A a given a e E C X, match the value of…

A: We need to check either a is there or not?

Q: If x is a limit point of a subset A of a metric space X, then sh

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Q: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

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Q: Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is…

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Q: 2. Let (X, A, μ) be a measure space. If {E} CA and E₁ E₂ E3 C..., then prove that H(UE) = lim (En).

A: Let X,A,μ be a measure space. Let {En}n=1∞⊂A and E1⊆E2⊆....then, To prove: μ∪n=1∞En=limn→∞μ(En).

Q: 6. Let E CX where X is a metric space. Show the limit points of E are the same as the limit points…

A: Limit Point: Let be a metric space. A point is a limit point of a set if for each there exists an…

Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

Q: Let1. (X, M,) be a space with measure. Show that for arbitrary A, B E Mthere is an equality: /x.du =…

A: We are given that (X,M,μ) be a measurable space and A,B∈M. We need to prove that  ∫AχBdμ=μ(A∩B).

Q: Let X and Y be normed linear spaces and F:X-Y. If for every Cauchy sequence {x, in X, the sequence…

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Q: Let (X, E) be a measurable space and E e E. If E1 e £ is such that E1 C E and ƒ is measurable on E…

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Q: 5. Let (X,T) be a discrete space, then Va E X, the set {a} is а. орen b. closed c. clopen d. is…

A: Our objective is to choose the correct choice

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let (S,≼) be a linearly ordered set with the Least-Upper-Bound Property. Let A and B be non-empty…

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Q: of E. 4. Let E CR be a bounded set. Prove sup E e Ë.

A: (4) Let E be a bounded subset of R  and let u be the sup E.  Then two cases arise-  Either u belongs…

Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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Q: a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.

A: (a) False f:E → INLet E= IR  Thenf: IR → INdefened as f(x) =  1  , ∨ x ∈ IRi.e There exist a…

Q: (b) Let (X, >, ) be a measure space. For A, Be ). Show that μ(AU B ) - μ(Α) + μ(Β) -μ(An ). %3D

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

Q: Let f : (X,Tx)→ (Y,TY) be a function,where TX is a topology on X and TY is a topology on Y. a) Show…

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

Q: Give an example of a continuous function T, and an open set O in a metric space (X, d) such that…

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

Q: A metric space is said to be separable if it contains a countable dense subset. (a) Is R separable?…

A: Set of all rational numbers Q belongs to R and Q is dense in R . Also set of all rational numbers Q…

Q: Let X be a metric space. Let A and B be separated sets in X. Show either E is a subset of A or E is…

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Q: Let AC R " be a measurable set and let a € R. Using the transformation theorem , show v ( a + A ) =…

A: This is a problem of measure theory.

Q: Let X and Y be two discrete spaces, then *

A: Note: Since you have posted multiple questions we are only allowed to solve the first one. Thankyou…