Let (X, d) be a metric space such that the intersection of open set and closed set is always open. Suppose that for each set E, there is a closed. set F such that FC E. Show that if there is a set J such that J is open and closed, then there is a sequence of set {Jn} such that J₁ J₂ J3 ... and Jn is open and closed, for all n E N.

Q: (c) Let X, Y and Z be sets. Prove that (X\Y)n (X \ Z) = X \ (YUZ).

A: We prove the given statement by using set theory.

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: Let A and B be sets. Prove that A U B = An B if and only if A = B

A: First show if A∪B=A∩B then A=B, then show that if A=B then we have A∪B=A∩B.

Q: Given the function f= {(n,n²)such thatn< 3,n€N} as a set of ordered pairs in set-builder notation,…

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Q: Let A and B be subsets of a space X such that A is an open set and B is a closed set. Then a. A-B is…

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Q: (c) Prove that a metric space is connected iff it contains exactly two sets that are both open and…

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Q: -prove that for any bounded Set E, There is abounded set A That is a countable intersection fopen…

A: The solution is:

Q: QLet (X,t) be a topological space and A,E are two subsets of x show that : 3. A is closed set iff A…

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Q: Which of the following statements is not true? (i) In any metric space (X,d) every closed ball is a…

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Q: Which sets are compact? The union of [0, 1] and [2, 3]

A: Concepts introduction: Theorem (Heine-Borel Theorem): A subset S of R is compact if and only if S is…

Q: 5. Let (X, d) be a metric space such that the intersection of open set and closed set is always…

A: As per the question we are given a metric space (X, d) space such that the intersection of open set…

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

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Q: List all the possible subsets of set G, where G = {-7,m,z}- %3D

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Q: Suppose A={-1, 0, 4}.Find a set B such that A is equivalent to B.

A: Two sets A and B are said to be equivalent to each other if both the sets A and B have same number…

Q: >) Recall that the symmetric difference of two sets X and Y, denoted XAY = (X-Y)U(Y-X), is the set…

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Q: (iv) Prove that for any sets A, B,C we have A U (B nc) = (AU B) n (AUC)

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Q: a and b be disjoint subsets of the universal set U. Assume n(U) = 110, n(A)=50 and n(B) = 25. Find…

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Q: Show that the interval [0,1] is a closed set in R

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Q: Q.No.2 Let X be a universal set and A, B be the subsets of X.Then prove the following. (a) ((A*…

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Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: prove: for any two sets X and Y, #(X x Y) = #(X) • #(Y)

A: To prove it equals the product of the cardinality of both the sets.

Q: Let R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f…

A: Given:

Q: Let (Z,t) be a cofinite topological space. The set {.-74,-72,-70,70,72,74,...} is

A: Given Z,τ be a co-finite topological space. Where, Z is the set of integers that is…

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

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Q: # 2. Prove that for all sets A and B, (A - B) N B = Ø. Proof (A Proof By Contradiction):

A: The given problem is to prove the following statement using the method of contradiction. To prove :…

Q: Let X and Y be two discrete spaces, then * X is never homeomorphic to Y X is homomorphic to Y if and…

A: Definition: Two descrete spaces with same cardinality are homeomorphic.

Q: Q1: (a) If (M, d) be a metric space and A M. Prove that A is the largest open set in M that…

A: The interior, or (open) kernel, of A is the set of all interior points of A: the union of all open…

Q: Prove that if XY, then X UZ YUZ for all sets X, Y, and Z.

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Q: Let A = {0, 1, 2} and let B = {-2, 0, 1, 2}. %3D

A: Given,        A=0, 1, 2, and B=-2, 0, 1, 2

Q: List all the ordered pairs in the relation R= {(a, b) | gcd(a,b) = 1} on the set {2, 3, 4, 5, 6}.

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