3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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A:

Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

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Q: 5. Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E) %3D

A: This is a problem of Real topology, metric topology.

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Q: 6. Show that the span of this set is all of R?. {}}

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Q: 18 What is the size of the space stretched by the set {x, x+x², x³, 1 , x²} ?    a) 2nd  B) 34…

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Q: 2. Sketch the following sets. Determine whether each set is open, closed, or neither and whether…

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Q: Consider RxR to be a universal set with subsets A, B, and C defined as follows. A = {(x,y)\x² +y°…

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Q: 2- Prove that H, is convex set.

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Q: let a be a nonempty compact subset of R and let B be a nonempty closed sub set of R such that A…

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A: Solution

Q: Give an example of a closed set in R ^ 3 with the usual topology that is not compact

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Q: 4. Use the definition of compactness to prove that a closed subset Y of a compact set X CR is…

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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO

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Q: Use the definition of compact set to prove that the set [0,1] is compact

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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: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

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Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: For any set A and B prove that: a) А - В %3D А - (ВПА) b) BC A' iff A n B = ¢ с) (А — В) U В %3D…

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Q: 3) let X be discrete topology and Ac X, find A ?

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Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

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: 4. Prove that the set Z is a closed set in R.

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Q: Given the set T := {(x, y) ∈ R^2 : |xy| Less or equ 1}. Is T a compact set? Show your working. If…

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Q: For nonempty sets A, B, and C, let f : A –> B and g : B –> C. Prove: If go f is one-to-one, then fis…

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Q: Let X be infinite set and x0 ∈ X. Consider the topology τ = {G ⊆ X : x0 ∈ G or G = ∅} on X. Is (X, τ…

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Q: 28

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Q: Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)}…

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Q: Let E be a set that bounded and c > 0. Prove that sup(cE) = -c inf(-E)

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Q: Let M be a closed subset in (X,d). Show that b(M) is a subset of b(b(M)) where b(M) is the boundary…

A: Given: M is a closed subset of X,d. To prove: bM is a subset of bbM, where bM is the boundary of M.