Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X. Prove that U",C, is compact.

Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: Let E1, E2, ..., En ⊂ X,d be compact subsets of a metric space X. Just using the definition of…

A: For a metric space we know that the for a compact set every open cover has a finite subcover. So…

Q: (a) Show (by calculation) that {u1(x), u2(x)} is an orthonormal set. (b) Find the projection p(x) of…

A: Given the space C0,1 of continuous function on the interval 0≤x≤1, with the inner product defined as…

Q: 2) let X and y be metric spaces and T. XY Y a ContinUOus mapping Then the image of a Compact subset…

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Q: A normed space X overR is separable if it has a conutable dense subset 0 صواب ihi

A: The given statement that " A normed space X over R is seperable if it has a countable dense subset "…

Q: Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of…

A: Consider, I=A⊂X : Ac is countable ∪ϕSince,Xc=ϕ is countable, X∈I Also given, ϕ∈I Assume, Aii∈N∈I…

Q: prove that an is bounded above.

A: In this question, we prove the sequence  an=(1+1n)n is convergent by given porcedure. i.e.…

Q: Let X be a disconnected space, that is X = UUV, where U and V are nonempty open subsets of X and…

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Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Let (X, d) be a separable metric space. into l∞.

A: This is a problem of functional analysis.

Q: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

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Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify tha the quotient…

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Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: Let ( Xr d be ame tric space,show that exist atopology on X induce b

A: Let (X, d) be a Metric Space We know that in a Metric Space  (1) arbitrary union of open sets is…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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Q: Prove that for any infinite index set J, the uniform topology on R° is strictly finer than the…

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Q: Let X be a discrete spaces then * X is homeomorphic to R if and only if X is finite X is…

A: The objective is to choose the correct option: Let X be a discrete space then  a) X is homeomorphic…

Q: If there exists open ball B (a, r) such that B (a, r) nG=Ø then a is not limit point of G

A: To check whether there exists open ball B(a, r) such that B(a, r) ∩ G= ∅, then whether a is not…

Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: Let X be a finite dimensional norm space. Then prove that M = {x € X |||x|| <1} is compact.

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Q: Show that every closed ball in R" is an n-dimensional manifold with Assuming the theorem on the…

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: Let X be an inner product space over C and x, y = X are such that || x + y || = || x-y ||, then show…

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Q: th atlas V (so that D

A: Given: manifold (M, A) To show: V (so that D(V) = A)

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: 1. Let M, d be a metric space with A C B C M. Suppose that B is totally bounded. Prove that A is…

A: Let (M, d) be a metric space and let A be a nonempty subset of M. The diameter of A is defined as…

Q: Let X be a topological space. Construct a topological space Px and a continuous surjection 0: X→ Px…

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Q: Let A CX with the discrete topology. Prove that ô(A) = Ø.

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Q: Suppose the collection = {B₁ (x): xe (0,1)}. Is B an open cover of the interval [0, 1) CR? Give an…

A: Given: The collection B=B12x : x∈[0, 1). To determine: Whether B is an open covering of the interval…

Q: Suppose p: X → Y is a surjective map from a topological space X to a set Y. Verify that the quotient…

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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: Let {Xa | a E A} be a collection of Hausdorff topological spaces. Prove that IαEA Xa is Hausdorff…

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Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function  f(x) =…

Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

Q: 1. Show that if & is a function from a nonempty compact metric space X to itself such that d(@(x),…

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Q: Let (X, d) be a metric space and let A be a non-empty subset of X. Prove that A is open if and only…

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Q: Let (X, || · ||) be a Banach space, J CR be bounded and F C C1 (J, X ). If sup ||f(x)|| < , f€F, rɛJ…

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Q: Let E,F ⊆ X,d be compact subsets of a metric space X. Just using the definition of compactness…

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Q: Show the following result: Theorem. Let (X, d) be a metric space, A C X, and let u: A → R be an…

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Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Let X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…

Q: If A is a compact subset of a metric space (X, d) and B is a closed subset of A, prove that B is…

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Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Show f is…

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Q: 1. Let X C R"-1 be bounded. Prove that X considered as a subset of R" is a volume zero set.

A: Given, X⊆ℝn-1 That is, X is a subset of ℝ in terms of exponential n. So, let we consider the set,…

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

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