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: Prove that in a metric space (S,d) every closed ball Sr[Xo] is a closed set

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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: Find a base for the Standord tapology-

A: This is a question of topology.

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: 7. Let X be a connected space. Let C be a connected subset of (X, T). If X\C=BUD where B and D are…

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Q: Suppose (S,d) is a metric space. How can we prove that S is open

A: To prove that a set S in a metric space (S,d) is open, we need to show that for any point x in S,…

Q: What is the standard error of the sample proportion who are​concerned?   b. What is the…

A: Here given that An employee survey conducted by a company two years ago found that 53​% of its…

Q: Give an example of a set X and topologies T1 and T2 on X such that T1 union T2 is not a topology on…

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Q: Let X = {a, b, c, d, e}. Create a σ-algebra X from set X. Explain why it is a σ-algebra.

A: Given : The given set is  X = {a, b, c, d, e} To find : We have to create a σ-algebra from the set…

Q: Consider a set X = {a,b} as a topological space with a discrete topology. Let J be an uncountable…

A: We will use basic knowledge of results and definitions of general topology.   We claim that Z is NOT…

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: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

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: Consider the set of elements {0, 1, ..., 56}. Define 57 blocks as {0+i, 1+i, 7+i, 19+i, 23+i, 44+i,…

A: Given: The set of elements Blocks: Define  blocks as  for (all computations to be done modulo…

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 R be the set of all real numbers with the euclidean topology. Define the set A = {(r, y) ER x R…

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

A: Consider the subsets [0, 1] and [3, 4] of R, the set of all real numbers with theeuclidean topology.…

Q: Show that the collection of all sets which are union of sets of the form [a, b) is a topology on R.

A: To Show: The collection of all sets which are the union of sets of the form [a, b) is a topology on…

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: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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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: (c) Prove that only Ø and R are open subsets of R

A: A subset Uof a metric space (M, d) is called open if, for any point x in U, there exists a real…

Q: bijective Show That There is map from CCA) to PCA).

A: Sol:- Let A be a set and let C(A) denote the set of all functions from A to {0,1}, which is the…

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: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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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…

Q: The closure of a compact subject of a regular space is compact.

A: