QUESTION 15 Let f. X Y be a continuous surjection between metric spaces. If X is compact then O A.X is complete O B. Y is connected. O C.Y is not compact. O D. Y is not neccessarily complete.

Q: Question 6. Let (X,7) be a topological space and A, BC X. Determine whether (A- B) CA -B or not.

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

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and…

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Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: Which sets are open in the metric space (IR, d), v i). [1, 2) ii). (1, 2) U (2, 3)

A: ii). (1, 2) U (2, 3)

Q: Let E be a subset of a space X. show titat cl(E) = EUd(E). %3D

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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: Let (X, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

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Q: Question 4. Let f:XY be a mapping from a topological space (X.r) to a topological space (Y.a). Show…

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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: Let fex) and g(x) be two (Piece - wi'se) comtinuous functions on Intervál caib]. Then if t) ¿gCx)…

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Q: 3. Prove that every connected subset of R is an interval.

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Q: Let X and Y be two T, spaces, then X×Y. :is

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Q: The set [0, 1] with the discrete metric is compact. True False

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Q: Let. f be q into medric Captinyous. marping of a medric rpare %.

A: Matric space

Q: Let X be the space of bounded and closed linear segments on the positive part of the real line, so…

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Q: 6 Prove that a closed set in the metric space (S, d) cither is nowhere dense in S or clse contains…

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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: Fix a prime number p. For x EQ - {0}, write x = p" () where a, b € Z, a and b are not divisible by…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If AnC = 4, BnC = ¢, µ(An B) = 0. Determine µ(A UBUC).…

A: Measure

Q: QUESTION 15 Let f. X Y be a continuous surjection between metric spaces. If X is compact then O AX…

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Q: Which of the following are compact spaces? (a) S" with a finite number of points removed. (b) The…

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Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

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Q: (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.

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Q: Which of the following sets is not closed on the real line with the usual metric? (i) The set of…

A: Option (iii) the set of rational numbers is not closed on the real line with the usual metric.

Q: IJAis dense in a metric space X. then A is closed True O False O

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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 (A)° = A – int A %3D…

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

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: Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition…

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

Q: Is the set S = {1,, 3, ...} with metric d(x, y) = |æ – y| for all x, y e S complete ? prove your…

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Q: Let x be an interior point of a subset S of a metric space (X,d). Show that x must also be a limit…

A: Interior Point: Let A be a subset of metric space X, d. A point x∈A is called an interior point of A…

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: Y is a 5. Suppose that X and Y are metric spaces and f X continuous onto function. Prove that if D…

A: Given, Suppose that X and Y are metric spaces and f: XY is a continuous onto function.   We have to…

Q: d(r, y) Let (X, d) be a metric space and let p(r, y) = for all r, y E X. Show that (X, d) 1+d(x, y)…

A: According to our company policy we are supposed to answer only first question i.e. (4) 4) in this…

Q: .2 Prove that in any metric space (S, d) every closed ball S,[xo] is a closed set.

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Q: Let A be a sSubset of a metric space A. Given r€A, define the distar from x to A to be dist(z, A) =…

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