(d).Show that the derived set E′E′ of any set EE is closed.

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

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Q: A point y in space either belongs to a given closed c ough y so that whole of the x lies in one open…

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Q: - Let (X, t) be a topological space and A, BCX, then 1- A° 2 A- b(A) 2- A° # A- b(A) 3- A° = A –…

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Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: Theorem 9. The following statements regarding the set E are equivalent: (i) E is measurable. (ii)…

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Q: Let A = psino a,+p a,. Evaluate A dl given that L is the path shown in the figure. 1. Verify your…

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Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and…

A: To prove: Let X, Y, and Z be topological spaces. A function  g : Z → X×Y is continuous if only if…

Q: In each of the following cases compute the set, L, 3) Let A C R. of limit points of A. (i) A=[2,3] U…

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Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{AC X : xo ¢ A} is a topology on X.…

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Q: Let T={a: a ER}, then T is Borel algebra.

A: Borel sets of real numbers are definable as follows. Given some set, S, a σ-algebra over S is a…

Q: Theorem 2.30. Let A be a subset of the topological space X, and let p be a point in X. If the set…

A: Given that A is a subset of the topological space X and p is a point in X. Consider a sequence…

Q: Let x E RP be a cluster point of a set E and that x ¢E. Prove that x is a boundary point of E.

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Q: Let lx, T) be a topological Space and A, B S X- if A is open, then show that An CIIB) s CLLANB). if…

A: Given X, τ be a topological space and A, B⊆X. Suppose, A is open then An arbitrary x∈A∩clB implies…

Q: A relation R defined on the given set A will satisfy

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Q: Let {An : n e N} be the indexed collection of sets defined by (2. 2-) 1 An Prove that: = Ø

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Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: Let G=(V, E) be a finite

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Q: Question # 01. Consder z= {acbos a-b =0; S2 Interpet that eguivale nce relation. ~ 18 is

A: That's easy, just show that this relation is reflexive, symmetric and transitive.

Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: Let E be a non-empty bounded subset of R. Prove that inf E is unique

A: Given: E is a non-empty bounded subset of ℝ. To show: inf E is unique.

Q: Does the follorin relations hip define y as a real furction of x Why," AN

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Q: Let Xbe a nonempty subset of R. Show that if c E R is a cluster point of X, then for any n EN, there…

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Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

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Q: 4) Determine whether or not each of the following subset of R with the euclidean topology is…

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Q: 4. Let (X, d) be a metric space, r E X, e > 0, and E = {y E X : d(x, y) < e}. Show that E is closed.

A: Let (X,d) be a metric space, with x∈X , ε&gt;0, and E={y∈X:d(x,y)≤ε} To show: E is closed. That is…

Q: Let Á = ry?a, + x³yzay, Evaluate A. dl along the path y = x2 from a(0,0,2) to b(3,9,2) Select one: O…

A: We will find out the required value.

Q: Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is continuous if and…

A: The given theorem is Let X,Y and Z be topological spaces. A function g : Z→X×Y is continuous if and…

Q: A topological space (X, t) is a T1-space iff t contains the cofinite topology on X.

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Q: of a space X and let x e X. Then x e A iff there exists a net in A (that is a net whi to x in X.

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Q: If S. F.dr is independent of the path C joining any two points, show that there exists a function O…

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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: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: 4) Determine whether or not each of the following subset of R with the euclidean topology is…

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Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

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: 2.3.11. Let d be the discrete metric on a set X. Explicitly describe the open balls B,(r) for all a…

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Q: 1) Let X be any non-empty set and xo E X. a) Prove that T = {X}U{ACX: 2o ¢ A} is a topology on X. b)…

A: Topology means we have to prove 3 conditions then we can tell that space is Topology.

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: The set E={xER:xs2x² – 1} with the usual metric is compact.

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Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: Let fELoi Show that i for any E>o, the set 3- S has a positive measure -

A: For f∈L∞ , f∞=infM&gt;0:μx:fx&gt;M=0For M&gt;0, defining the set AfM=X:1 fx&gt;M⇒For f∈L∞,…

Q: (D), (a) Let E = Q. Find the set of cluster points of E and prove that E is neither open %3D nor…

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Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: Let E ⊆ R. Show that E is measurable if and only if χE is measurable

A: