Q: In usual topological space (R.Tu). answer the following: 1- Is [0,1] is compact set in R? Why? 2-Is [0,1] in connected set in R? Why? 3- Is [0,1] is path in R? Why?

Q: Let U = R be the universal set and A = [1,3], B = (-∞,6], C = (-4,0). a) AUB b) BnC c) A-C d) BC

A: Let  be the universal set and  Then we have to answer the following...

Q: Prove that for any sets A, B, C: (AUB)-B A - B

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Q: 4- A subset [a, b] R is compact set in: a- (R, Td), b- (R, T), c-Both (R, Tu) and (R, T₁) d- No one.…

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: 4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And…

A: Method of contradiction : suppose A is disconnected ,then there are two disjoint closed non empty…

Q: 13. The interior of a set coincides with the interior of its closure..

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

Q: 9. Show that the property of "having nonempty open sets with a finite number of points" is a…

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Q: 17. Prove: S' is a closed set.

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Q: (E, Q4: state and prove L is a normed space ? e a ne. Vila a convex set.

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Q: Prove that in R with the usual topology, the only open sets connected by paths are the open…

A: the solution is given below

Q: 3. Let R be the universal set and D = (-7,4], E = (0,8), F = {r: r ≤ 10, r € R} be the subsets of X.…

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Q: 13) Let A = {2, 3, 4, 5, 6, 7, 8} and define a relation R on A as follows: for all x, y € A, хRy+3|…

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Q: 3) Consider the subsets [0, 1] and [3,4] of R, the set of all real numbers with the euclidean…

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Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

Q: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

Q: a) E = IN x IN; (a, b) R (c, d) → a<c

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Q: tify rigorously the given set is closed in R³. A = {(x, y, z) | x ≥ 0, y ≥ 0, z ≥ 0, x+y+z ≤ 12}. B…

A: We can prove by showing complement of a set is open.

Q: 4. Prove that the intersection of an arbitrary collection of compact sets in R is compact.

A: (Heine-Borel Theorem). A subset A of R is compact if and only if A is bounded and closed (with…

Q: Please provide a succinct explanation below (graphs and calculations may be necessary): The set ECR…

A: As per the question we are given two different sets as : E ∈ R2 : x + y = 1 (a straight line on the…

Q: nsider the sets A = {1, 2} and B = {3,4}. Verify that, for these sets, the following iden d: 1.…

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Q: Both questions below refer to set A = Z – {-2,0}. a) Give a partition of A into infinite many…

A: Solution: Definition: Let Ψ be a partion of the set A. That means, Ψ is the collection of subsets of…

Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: 8. Give an example of each of the following : (a) a set which is a nbd. of each of its points, (b) a…

A: (b) Given the example of a set which is not a nbd. of any of its points

Q: Let Gbe an open set in R". Two points x, Y e G are said to be equivalent if they can be joined by a…

A: Since every open subset of Rn can be written of countable union of disjoint open balls and each ball…

Q: If D(L AOB) states the set of points located on the OA side containing point B and the OB side…

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Q: Justify rigorously the given set is closed in R³. (a) A = {(x, y, z) | x ≥ 0, y ≥ 0, z≥0, x+y+z≤…

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Q: 5. Describe the closed sets in the following topological spaces: a) (X, E,) wherepe X. b) (X, P,)…

A: (a) Given,           (X, Ep), where p∈X

Q: 11. Show that the set of real numbers with the usual topology is not homeomorphic to the R set of…

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Q: -4 2. -5

A: Let us consider matrix A of order n×n. Null space of matrix A is given equation by equation AX=0.…

Q: Are the following true or false? If true, prove. If false, give a counterexample. (a) The…

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Q: ☺ Let (u,Ue...,un) be a lineorly ndependent list. Which True Statement? is a has --- a non-zero…

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