3. Show that every locally compact space is compactly generated.

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: c. Suppose that G,H < Sym(Q). Show that the corresponding spaces (2,G) and (Q,H) are equivalent if…

A: Suppose that G, H ≤ Sym (Ω). Let G, H be a multiplicative group and let Ω be a set. An action of G,…

Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

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Q: Prove (Lo (x),|- ||∞) is a normed space

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Q: Define the closure of a sot I in a motric spack. How is the boundry it a sut I related to it clogwai

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Q: 11. Let (X, d) be a pseudometric space, and suppose d has the property that d(a, b) >0 whenever ab.…

A: 11) To prove that every finite subset of X is closed, we need to show that its complement is open.…

Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

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Q: 10. Let (X, 71) be a Lindelöf space and (Y, T2) be a compact space. Then show that the carte- sian…

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Q: (b) Show that every compact set of T₂-space is closed set but the convers need not to true. (c) Let…

A: T2-space: A topological space X is said to be T2-space for given x, y∈X, there exists open sets U…

Q: ) Prove the theorem that every metric space has a completion

A: Every metric has a completion.

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: A completely regular space is also T1-space 1 True 2 False

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Q: Given the trapezoid below find the measure of GH.

A: Given,

Q: Let X be a topological space

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Q: 1) Every convex set of a vector space X is affine set.

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Q: 2. Define the concepts of (a) first countable space; (b) second countable space; (c) separable space…

A: According to Bartleby guidelines I have explained defination of first countable space, second…

Q: let (x,t) be a topological space prove that (x,t) is not connected if and only if there exist A,B…

A: Let (X,t)  be  not  connected.We  have  to  show  that  X=A∪B ,where  A and B are  elements  in t…

Q: (a) Let || • || 1, || • Il2 be norms on a linear space X and let e > 0, 8 > 0 be fixed. Find…

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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: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: Q7) (a) Let X be a Banach space and R the usual normed space. Show that ||(z, a)|| = ||||+|al is a…

A: (a) Given a Banach space  and  the usual normed space, we want to show that the norm  is complete on…

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: Show that: All completions of a metric space are isometric.

A: According to the given information, it is required to show that all the completions of a metric…

Q: 2.8.12 Let X be a metric space. Prove that an arbitrary intersection of compact subsets of X is…

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Q: (i) Show that a subset A of a topological space X is closed if and only A = A (ii) Prove that A…

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Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: 2.8.15) Prove that a compact metric space is both complete and separable.

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Q: ctors - Z. S. Madhi

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Q: (a) Prove that the metric space (2,11) is complete. (b) Show that the metric space (Q.H) is not…

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Q: Prove or Disprove the following: a) The product Space of two Connected spaces must be connected.…

A: As there are two proofs asked, I will solve only the first one. Let M1 and M2 be connected spaces.…

Q: Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a…

A: 1)  Consider X is a T2 - space. To prove that, X is a T1- space. By the definition of T2- space,…

Q: (e) Show that a compact metric space can be covered by finitely many balls of any given radius. (f)…

A: A compact space has the property that every open cover has a finite subcover. In the context of…

Q: .. .): P, is a polynomial space with degree with order n and less than equal n. Is the space S =…

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Q: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

A: Given topological space X such that X contains an infinite closed discrete subspace. We now show…

Q: Check that the following are vector spaces by proving in full detail that all axioms (one by one) in…

A: 1. a)      Let u, v, w ∈Rn ; so u= (u1,u2,u3,......,un) ,v=(v1,v2,v3,........,vn), w=…

Q: If A, B are subsets of the topological space X show that Ext(AUB) = Ext(A)n Ext(B).

A: Given: A and B are subsets of the topological space X. To Prove: ExtA∪B=ExtA∩ExtB

Q: Corollary 2.53 Let (M, d) be a metric space and K C M. The following statements are equivalent: (i)…

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

Q: Give an example that satisfies in arbitrary metric space, boundedness and closedness do not…

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Q: (2i=1 αix Zi=1 f(Σ+1 aix; ) = Σ+1 af(x;) * True False Any finite dimension vector space with dim(X)…

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Q: (a) Prove that every closed subset S of a compact metric space (M,d) is compact in M.

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

Q: Determine the dimensions of the following spaces:    a) span ((1, 2, 3), (2, 4, 6), (3, 6, 9))…

A: In this question, we will find the dimensions of the spaces.

Q: ": Prove that ( R", ||||) is complete metric space.

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Q: a Normed &pace and and CL(X,Y) denste, of bounded Let X,Y be a Suppose BL(X,Y) respectively, the…

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Q: Every T2 (Hausdorf f) space is T1 space. True O False O

A: Since T2 is a product preserving topological property. So T2 space is a T1 space.