Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: 3. Let (X, d) be a metric space and x1, x2, ... , Xn, ... a sequenc

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Q: Show that R is a Banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: ℝ3=x(1, x(2), x(3)) :…

Q: Let (X,T) be a toplogical space, A, BC X. Show that АС В Ас в, АсВ. 介

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Q: Let X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

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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: A ° = A – JA

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

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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: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Let X be a topological space

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Q: let and be nonempty metric Space x E X. Ihen A B Sebsets of and let d(x, A), dlA, B) and dlA) are…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: Every metric space is a hausdorff space .

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Q: 3. Let (X, T) be the topological space defined in question 2., then the closure of {2} is a. X b.…

A: It depends on the given topological space. Here topological space is not specified. If it's usual…

Q: (a). Define three different metrices d1, d2, d3 and their induced norms on R³. Also prove the…

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Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Let X and Y be two topological spaces. X is connected and Y is disconnected are they homotopy…

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Q: = {a, b, c, d}, t = {Ø, X, {b, c}}. Then in the topological space (X, t), {a} = %3D %3D %3D {a, b}…

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Q: Construct examples to show that if X and Y are topological spaces with G acting on them such that…

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Q: Prove that every finite T,- space has the discrete topology.

A: T=(X, τ)  (X, τ) is a T1 space if and only if all points of X are closed in T.   Let X≠∅ be a set.…

Q: Let X = {a,b,c} prove that topological space on the set X T = {X. Ø, {b}, {a,b}} be a

A: Let X={a,b,c} be a set. To show that T={X,ϕ,{b},{a,b}} be a topological space on the set X.

Q: If S, S, are two subsets of an inner product space V(F) then show that

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Q: Exercise 2.27. Show that a set U is open in a topological space X if and only if every point of U is…

A: Let, U is an open set in a topological space X. Then, U⊂U  So, U⊂int(U) and int(U)⊂U Hence, U=int(U)

Q: Show that R is a bianch space.

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Q: Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A…

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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 C be a compact subset in a Hausdorff topological space (X,T) and b∈X a point. If b∉C then prove…

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Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: if

A: R is not connected if T is the finite closed topology.

Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: * Let R be with the co-finite topology. If A = {1,3,5, 7,..}, then Aº R O Q O A O N O

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Q: A of a metric space (x, d.)

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Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

Q: Suppose that {u, v} is a linearly independent set. Show that the set {u – v, u + v} is also linearly…

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Q: (ii) A topological space (X, T) is said to have the fixed point property if every continuous mapping…

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

Q: (d) Every continuous map on any metric space is an open map.

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

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Q: Let Z be with the co-finite topology If A = {1,3,5, 7,.. }, then A A O O O

A: As Z is with co-finite topology. Every open neighborhood of an infinite set must contain infinitely…

Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: Let X and Y be two discrete spaces, then * X is homeomorphic to Y if and only if X and Y have the…

A: X and Y are two discrete spaces then X and Y   The topology is  discrete then for x∈X  is an open…

Q: Show R³ is a banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: To show:  is a Banach…

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: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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

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Q: Theorem 8.11. For topological spaces X and Y, X ×Y is connected if and only if each of X and Y is…

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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: 2. Let (X, 1) be any topological space and (X, Tf) be the finite complement topological space. Show…

A: Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show…