Show that is not compact for the Standard topology but is compact for the cofinite topology.

Q: Let τ and τ' be two topologies of a set X. Is the family τ∪τ' formed by the open elements of both…

A: Union of two topologies is not necessary a topology.

Q: Show that every metric space is an Hausdorff space.

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Q: [Falkner Section 11 Exercise 26] Let A, B, and C be sets. Prove that if f: A → B and g: B→ C are…

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Q: tinuous. 1) Show that a Lipschitz map between metric spaces is uniformly con-

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Q: Prove that an infinite subset of a compact space has a limit point.

A: We can Prove an infinite subset of a compact space has a limit point.

Q: Prove that B = (0,1) is NOT a compact set.

A: To prove that (0,1) is not a compact set.

Q: Suppose (X, d) is a nonempty metric space with a discrete topology. Show that X is connected if and…

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Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

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Q: Suppose that Q : R" → R is a quadratic form attaining the minimum value on the set ||x|| = 1. %3D…

A: Given that                           Q : ℝn→ℝ is a quadratic form attaining the minimum value…

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: 5) Let A= (2,3) U {4} be a subset of R with the standard opology. Find Ə(A).

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Q: Show that every normed liner space which is linearly isometric with a separable space is separable.

A: A normed linear space or normed vector space is a vector space over the real or complex numbers,…

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: ► Exercise 11.2. Prove Proposition (c) A finite product of covering maps is a covering map.

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Q: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Can you show an example of Null space?

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Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: 22. Let v₁ = (1, 6, 4), v₂ = (2, 4, -1), V3 = (-1, 2, 5), and W₁ = (1, -2,-5), w₂= (0, 8, 9). Use…

A: Given vectors are Here observed that So we have each of .In other words ,

Q: What is the Boundedness Theorem?

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Q: Define Planar Nonplanar  Kuratowski’s Theorem Complement

A: Graph: G=V, E is called graph where V=Set of vertices or nodes E=Set of edges Complete Graph: A…