Let f: X →Y be a continuous surjection between metric spaces. If X is compact then ............................. A. X is complete . B. Y is connected. C. Y is not compact. D. Y is not necessarily complete.

Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: Let X and Y be topological spaces and let f : X →→ Y be a function. Prove that the following…

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Q: Let f: X Y be a continuous surjection between metric spaces. If X is compact then…

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Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = A O (Ā)ª = Ā - int A |…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: A. Let H be the set of all points (x, y) in R² such that x² + xy + 3y = 3. Show that H is a closed…

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Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: I have proved all three conditions gor norm

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: For the line that has the equation X1 + 2X2 = 10, will pass through: %3D O (5,4) (10,1) O (2,4) O…

A: Given that  The equation of a line is x1+2x2=10 To find a point that will pass through the given…

Q: 4. Suppose X and Y are normed spaces, dim Y <0, f:X→Y is linear and f(x) = Y. If ker(f) is closed.…

A: To prove that ƒ is continuous, we need to show that for any given ε &gt; 0, there exists a δ &gt; 0…

Q: Q.Let f(x): [a,b] →R be a bounded mapping and we find a partition P satisfies U(P, f) – L(P1, f) < 2…

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Q: Let (X.d) be a metric space, x ∈ X, epsilon > 0, and E = { y∈ X: d(x,y) ≤ epsilon}. Show that E is…

A: A set E is closed in X if it contains all of its limit points.

Q: prove the following claims: suppose that ƒ is a convex differentiable function, then for any x, ye R…

A: To prove: Suppose f is a differentiable function then for any x,y∈ℝn, the following inequality…

Q: Let X = N and d be the usual metric, d (x, y) = |x – yl then d can induces a norm on X True O False…

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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: 3. Let d(x, y) := |2 - 2| for x, y E R. . is a metric on R. Show that (R, d) is not a complete…

A: Cauchy sequence: A sequence x1, x2, x3, ... in a metric space (X, d) is called Cauchy if for every…

Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnC = ¢, BnC = ¢, µ(An B) = 0. Determine µ(AUBUC).

A: Measure theory

Q: Let (X, || ||x) and (Y, ||- ||y) be the normed spaces. Let XOY := {(x, y): xe X;y ≤ Y} denote the…

A: We know that a norm  on a vector space  is a real valued function on , which has following…

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y & R: y = []; x E R) then fined the following…

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Q: Let (X, B(R), ft) be a Borel measure space. ANC= 6 BOC =0 p(AnB) = 0. Determine ju(AUBUC) %3D

A: Since sets are disjoint so measure will be the sum

Q: Let X = Q be a metric space with d(x, y) =r - yl, for any z, y EQ. Define the set E = {r € Q: 2 <³…

A: Since for any , we have , since otherwise , which contradicts to the definition of E. Similarly, we…

Q: 1. Let X and Y be topological spaces. Prove that a function f: X → Y is continuous if and only if…

A: Theorem used: Let X and Y are topological spaces. The function f:X→Y is continuous if and only if…

Q: Let A be a non-empty subset of the metric space (X,d). show that |d(x, A)- d(y, A)| < d(x, y)For all…

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Q: 2.2.15. Let X be a metric space, and suppose that xn → I and yn → K. Prove that d(xn, Yn) – + y in…

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Q: Q. Consider the measure space (R, M(R), µ), if x € Q 1 -2 let k(x) = { , h(x) = -2,then %3D ¢ Q if x…

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Q: Q3: state whether the following statement are True or False : X 1. The set A = {x ER: x² -5x + 6 ≤…

A: To answer question (1) the given questions recall the definitions of Maximum , infimum and maximal…

Q: 8. Let (X, d) be a metric space, and A, B are subset of X, then (i) Fr(A) = An (A)° = A – int A (ii)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: (a) For the following subsets in an indicated metric space determine the interior, boundary points…

A: Interior Points: Let (R,d) be a metric space and S⊆R. A point p∈S is called the interior point of S…

Q: ! 0. JUUTTUU M 10. Compute the distance between the functions f(x) = 5 + x and g(x) = 6 + 3x in…

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Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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Q: . Let X be a nonempty set with metric d. Show that the function e defined by e(x, y) = min[1, d(x,…

A: We know that d(x,y) is a metric on X. Hence d satisfies all the properties of a metric.  Now e(x,y)…

Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

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Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

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Q: D = {2 € 0³/2-1+i) 2}

A: Here we need to find boundary. So we just need to change less than sign to equal sign.

Q: Let (X. d) be a metric space and define p: XxX by ptr.g)= dx.y) Prove that (X,p) is a metric…

A: Introduction: To prove that (X, p) is a metric space, we need to show that the following three…

Q: QUESTION 11 then f(X) Let f X→ Y be a continuous function between metric spaces. If... subset of y.…

A: Let f : X to Y be a continous function between metric spaces. Then, we have to determine under which…