Theorem (Axioms of a Closure): Let (M,d) be a metric space and let S,T C M. Then (1) 0 = 0 and M = M. (2) If S ST, then ŠSŤ. (3) Š is the smallest closed set in M such that S CS. (4) S is closed in Me S = S. (5) 5 = 5. (6) SUT = SUT. (7) In general, (S n T) S §nT. But, (SOT) # $ nT. (8) S = Fe. %3D

Q: Ex: Let (X, d) be a metric space and E, andE2 EX Show that whether each one of the following…

A: According to guidelines I solve first 3 subparts so kindly request you to repost the remaining…

Q: (1) Let X be an infinite set and set r to be the set given by T = {0} U {U C X : U° finite}. (a)…

A: τ={ϕ}∪{U⊂X :Uc is finite }  (a) Xc=ϕ, which is finite . so X∈τ let A,B∈τ Then Ac,Bcare finite…

Q: (1) Ø° = Ø and M° = M. (2) If SST, then S° ST°. (4) S is open if, and only if, S = S

A: As per the company rule, we are supposed to solve the first three subparts of a multi-parts problem.…

Q: (X,e) Let be a metric space The diky-k.glry (KEIR) function defined from XxX to IR is a metric on X,…

A: ρ(x,y) be a metric on X ,then it satisfy all the properties of metric  now d(x, y)=kρ(x,y)   k∈R…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Q7. Assume that (R,T) is the usual Euclidean topological space. N = {1,2, 3, ....} C R. Determine…

A: There is no interior point of N.

Q: Prove the following: Suppose that {X; : i e I} and {Y; : j € J} are indexed families of topological…

A: let us define ϕ:∏i∈IXi→∏j∈JYj=∏i∈IXα(i) by ϕx1,x2,x3....=xα(1),xα2,..... For any x=x1,x2,.... and…

Q: 3. Do the following task. a. () State formally what do we mean by being a metric space of X with…

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Q: Let X and Y be normed spaces. Then show that the following hold : (i) If Y is a Banach space, then…

A: We consider an arbitrary Cauchy sequence (Tn)n≥1 in BL(X, Y) and show that (Tn)n≥1 converges to an…

Q: (a) For the following subsets in an indicated metrie space detennine the interior, boundary points…

A: See the attachment.

Q: d set): Let (M ,d) be a (SUT)' = S' U T'.

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Q: Q4. Let (X,T) be a topological space where X = {m,n, q, r} and T = {4, X, {q}, {r}, {q,r}, {m, q,…

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Q: Q8. Assume that (R²,T) is the usual Euclidean topological space. D = {(x, y) E R², y = x} C R².…

A: Given below the answer in detailed

Q: Material

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Q: let Qid) be a metric space and AiB Bubeet of x ACB = d(A) cd(B) [ Soi d(A) is closed set] P B is…

A: Let X,d be a metric space and A, B⊆X.   We will use the following in the proofs, Definition of clA:…

Q: 2) Give an example of a metric space such that the closed ball Bå(x,r) is not an open set for any x…

A: To Find: Give an example of a metric space such that a closed ball B¯d(x,r) is not open set for any…

Q: (7) Which of the following is disconnected space: (a) (R\ {0}, Tusuat) (b) (R, Tusuat) (c) (X, t).…

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Q: Ans 7 only

A: To establish the required topological properties related to interior and exterior points.

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: Given: The statements: a) X is a Hausdorff space; b)  The diagonal D=x, x: x∈X is closed in X×X. To…

Q: M is an infinite set. If x,y ∈ M, explain the relation f below            {  1 , if x = y   f(x,y)…

A: We have to solve given problem:

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: Homworke 1.1 Prove that the Euclidean space R" is a metric space under the n Euclidean metric d(x,…

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Q: Let (sd) be and suppose is defined by a metric space that P: SXS - PR p(x,y) = d(x,y) i + dix₂y) For…

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Q: pls explain to me step by step. pls write clearly and dont skip any steps. thanks

A: To test the contraction mapping property for the same operator T in two different settings . (C[0,1]…

Q: Let V = F" where FE {C, R} show that V is a Banach space with the norm ||x| = maxisisn{xil VxEF".

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Q: Show that the closed ball Y in a metsic SPace (X,d)is a closed set. Also, show that it (x,d) is…

A: Show that the closed ball Y in a metric space (X,d) is a closed set. also show that if (X,d) is…

Q: = 1. Let (X, d) be any metric space. Difine p: X X X → R as p(x, y) d(ry). Then (X, p) is a metric…

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Q: Pls explain to me step by step and pls dont skip any step. Thanks

A: To prove that the taxicab distance is actually a metric on the plane

Q: If A is a subset of a metric space X, then show that A = (x: each neighbourhood of x intersects A}

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Q: Find all limit points of a set A = {(2+,2 – 1)|n e N} on the Euclidean plane with the usual metric.…

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Q: Theorem (Axioms of a Derived set): Let (M,d) be a metric space and let S,T C M. Then (1) S CT S' S…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: (c) let t be indiscrete topology on set of integers Z, show that whether (Z, t) is regular space or…

A: Given : ℤ , τ is indiscrete topological space. To show : To show whether ℤ , τ is regular , T0, T2,…

Q: Theorem (Axioms of an interior): Let (M, d) be a metric space and S,T CM. Then: (1) 0 =0 and M = M.…

A: Note: Our guidelines we are supposed to answer only three subpart . Kindly repost other subpart as…

Q: (1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a…

A: Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this…

Q: Theorem (Axioms of an interior): Let (M, d) be a metric space and S, T M. Then (5) S°° = S°.

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Q: For each of the sets find the set of boundary points and the set of interior points. Also state if…

A: We have, (a) [0,1]U{4} (b) x,y∈ℝ2:x2+y2=1 Here we have to find the  o find: Interior Boundary…

Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: The following statements about a topological sapce are equivalent: (a) X is a Hausdorff space; (b)…

A: We have to prove that the following statements are equivalent : (a) X is Hausdorff space (b) The…

Q: pls explain to me step by step and pls write clearly. thanks

A: To verify that the contraction mapping condition is NOT satisfied using the given functions f(t) and…

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) =…

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