Let T_us be the usual topology on R and T_| be the lower limit topology generated by the unions of {]a,b]/ a,beR;a

Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Exercise 8. Any map f: X Y between topological spaces which is continuous at x = X is sequentially…

A: Continuity and sequential continuity are defined below

Q: Suppose Y=[ 0 ,2 ) . In this case A=[1,2 ) is not closed in X; while closed in Y. In addition, G=[0…

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Q: lim In (z\x2 + y²) P - (5, 5, 1)

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Q: Consider a linear map T : R″ → R" with the property that null(T) = range(T). (a) For n = = 2,…

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Q: Let f: X->Y and g:Y->X be maps between sets. Suppose that g ∘ f is equal to the identity map id_X:…

A: Given f:X→Y and g:Y→X be the maps between the sets.  Suppose that g∘f=idx. Then We need to prove or…

Q: Consider the Banach space C[0,1] of continuous functions on the interval [0,1] equipped with the…

A: The Banach space C[0,1] is the space of all continuous functions defined on the closed interval…

Q: F onto a subfield of an algebraic closure F Theory 9. Let Z2 be an algebraic closure of Z2, and let…

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Let ƒ: R² → R² be an isomorphism where ([-])-[²] and f([:]) = [] = Find f ¹ƒ( [7¹]) X 0 2 2 0

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Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: Let X = R. Define an equivalence relation ~ on X by x ~ y if %3D and only if x – y is an integer.…

A: Let X, 𝛕X be a topological space, and let ~ be an equivalence relation on X. The quotient set,…

Q: 1. Given a set X and a subset A CX, define a function ƒÃ : X → {0, 1} by setting f₁(x) 1 if ¤ ¤ A…

A: (a) fA : X → {0, 1} is defined as             fA(x) = 1 when x ∈ A                       = 0 when x…

Q: Let: f : R → R with f(x) = 2x ^2 + 1. Prove or disprove that f is injective (one-to-one).

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Q: Let f: R → R such that f (x) = x ^ 2 Analyze if the function is continuous by taking the domain and…

A: Continuous function- Let X and Y be topological spaces. A mapping f:X→Y is said to be continuous if…

Q: Lot X= [a,b] with the standard topology and let Y= [9₁6) with the Cofinito topology. Why is the…

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Q: Let w(z) be a continuous pomitive function in a bounded domain in . Define Verify is an inner…

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Q: If m, n ∈ N, then (f^m)^n = f^(mn).

A: To prove two functions are equal, we need to show that both the functions have same function values…

Q: Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f…

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Q: 5) Prove that, for any set X, there does not exist a surjective (onto) mapping from X to P (X).…

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