Let x be O and all subsels f X whose complements Prove that (X, z) infimite set and let z consist of are cauntable. an (X, Z) is topological space

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: 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: Let (X,T) be a separable topological space, let A be an open subset of X. Then (A, T,) is a…

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Q: Prove that if f is a continuous mapping of a compact metric space, X, into a metric space Y, then…

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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: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

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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: 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: Let

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Q: Let X be an uncountable set and let oo EX be fixed. Let TX = {U E P(X); 0$U or X- U is finite} Let Y…

A: Given: Let us consider the given X to be unaccountable and let ∞∈X be fixed.

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: Suppose (X,S,u) is a measure space and f,h: X → F are S-measurable Prove that I|fh||, < ||f||, ||4|a…

A: Given X,S,μ is a measure space and the functions f,h:X→F are S-measurable. Suppose 1p+1q=1r,…

Q: let C be a compact subset Tn a Haus dorft topolopfcal space (K,R) and bex a If bac then prove that…

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Q: Any finite dimension vector space with dim(X) is equal to n is isomorphism with F dows but O True O…

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

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: 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: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

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: Letxandy and z be normed space s and s: xy andTIYz bounded oper atorsprove

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Q: Let X= {A ⊆ ℝ| 1 is not in A and ℝ \ A is finite}. Prove that ℝ is a compact Hausdorff space with…

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Q: If S be a n-dimen Sional normed Space, then prove that its dual space is al so n- dimen Sional .

A: A normed space E is called a Banach space if it is complete, i.e., if every Cauchy sequence is…

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: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: Let R be equipped with the standard topology. Define q : R? → R, q((x1, x2) : Jx1, if x1 2 x2 |X2,…

A: Question: The function q:ℝ2→ℝ defined by qx1,x2=x1, if x1≥x2x2, if x1&lt;x2 where ℝ2 and ℝ are…

Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

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: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

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: Let X be a set with more than one element. (X,7) is a topological space such that Vxe X, {x} €T then…

A: Given the set X contains more than one element.

Q: Let V be an inner product space over F. (a) Let u, w e V. Prove that ||2+ (u, w)w|> |u+||(u, w)w|2,…

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Q: Let M, be set of all rare subsets of a metric space X. Prove that countable union of M, is rare.

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

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

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: 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) and (Y, T1) be topological spaces and f a mapping of X into Y. If (X, T) is a discrete…

A: For proving f is continuous enough to prove that inverse image of every open set is open.

Q: Assume X is Hausdorff. If K C X is compact and x ¢ K, show that there exist disjoint open sets U and…

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Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

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Q: (2) Let (X,r)be a topological space, If for every A open set, and x E A", 3 Ux and Vy open sets,…

A: A topological space X is Hausdorff if for each pair of distinct points p and q in X , there are…

Q: Every inner product space is a metric space O True O False

A: TRUE

Q: Q6: (a) In any metric space (S, d), prove that every compact subset T is complete. (b) Prove that…

A: A metric space is a set together with a metric on the set. The metric is a function that defines a…

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 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: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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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,…