Let B be the collection of all Borel sets in R.. For any such Borel set U, define B U as the collection of all intersections between U and the members of B, that is B U= {BU: Be B}. Show that B U is a o-algebra in U.

Q: Show that every countable subset of R" has Hausdorff dimension 0.

A: claim- to show that every countable subset of ℝn has Hausdorff dimension 0. it is known that…

Q: Let A be an open subset of R. Is it true that (A)° = A?

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Q: Let C, D be subsets of a set U. By proving two inclusions, show that (CC n D) U (C n D) = D where Ce…

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Q: Let A1,·.., Am be compact subsets of R. Prove that Aı × · … × Am is a compact subset of Rm.

A: Given, A1,A2, ... , Am are compact subsets of R. We prove A1x A2 x ... x Am is a compact subsets of…

Q: A – Ö B, = Ü(A – B.). Ü(A – B;). i=1

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Q: Let Y be an infinite subset of a compact set X ⊂ R. Prove Y' does not equal ∅.

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Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: Consider the set X = {a, b, c, d} with the discrete topology, construct a basis of space that has no…

A: Consider the set X=a,b,c,d with discrete topology. Consider the basis of this set as follows.…

Q: Let C, D be subsets of a set U. By proving two inclusions, show that (Cºn D) U (C N D) = D where CC…

A: Identities of the sets

Q: sketch the set {z ∈ C : Re(z) = Im(z + 2 + i)}

A: We have to sketch the set {z∈ℂ:Re(z)=Im(z+2+i)} Let z=x+iy where x,y∈ℝ

Q: Let (X, d) be a metric space. a) Assume that A is a (finite or infinite) collection of open sets.…

A: To orove the required topological properties of open sets in a metric space (X,d)

Q: Let (X.J) be a topolgical space and {C1, C2 Cn} be a finite collection of compact subsets of X.…

A: Given Let (X,I) be a topological space and C1,C2,......Cn be a finite collection of compact subset…

Q: 6. Prove that if x1, x2,..., Xn are distinct points in a Hausdorff space X. Prove that there are…

A: Let X be a Hausdorff space. Then in X any two distinct points are separated by disjoint open sets.…

Q: Let X be a set equipped with the finite complement topology show X is compact Show that X is…

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Q: Let Y = {(x, y) = R² | x ‡ 1 and y ‡ 1} be a subset of R². Show that Y is an open set in the…

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Q: a. Find a bijection between Zt and the set {1,1/2,1/4, 1/8, 1/16, 1/32, ...}. b. Show that if A is…

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Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: Write down four distinct elements of the set {z ∈ C : z − z ̄ = 2i}.

A: The given set is z∈ℂ:z-z=2i We have to find the four distinct element of the given set.

Q: жат fuss vodo у льа изба и и & Dis PAS

A: To prove that an open ball in a metric space is an open set. Using the definition: Consider set A…