Let A, B and C be the subsets of U. Then prove that (A UB) -C = (A – C)U(B – C) by using laws of sets |

Q: Exercise 16. Prove the above theorem. Hint: show A ∪ B = A ∪ B − (A ∩ B). Theorem 35…

A: To prove: A∪B=A+B-A∩B

Q: j) A U A = A k) A U $ = A

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Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

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Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: (2) Show that if f has support in a compact set A CR and g has support in a compact set B C R then f…

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

A: To check: Whether the statement is true or false: Given: The statement is "The collection of all…

Q: For any set S c R, let 5 denote the intersection of all the closed sets containing S. (a) Prove that…

A: Consider the given information.

Q: Let A = {AC R| A is countable} and Az = {A c R| A° is countable} which is a measurable space? (R,…

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Q: prove for every two sets A and B that A=(A-B)U(A n B)

A: To prove: We want to prove  for every two sets A and B that A=A-B∪A∩B

Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let X be an ordered set, and let A be a subset of X.Consider A as an ordered set, where the order on…

A: Given that X be an ordered set and let A⊂X, A is an ordered set. Y⊂X and Y is convex  in X. The…

Q: 3. (a) Prove that A\(BnC) = (A\B)U (A\C) for any sets A, B, C

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Show that if A and B are bounded subsets of IR, then inf(A U B) = min{inf A.inf B}. %3D

A: To show  If A and B are bounded subsets of R then inf(AUB)=min{infA .infB}

Q: Prove that if S is a bounded set then S is bounded.

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Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

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: 1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X,…

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Q: 7. Let X be a set with |X| = n € Z4. Prove that for any set Y, |Y| = n if and only if there exists a…

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Q: Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact. * O…

A: "Let X be an infinite set endowed with the finite complement topology T, then (X,T) is compact." The…

Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B) n (A x C)

A: we have to prove the lemma .

Q: A∩((X−B)∩(X−C))=∅

A: We have to prove that A∩((X−B)∩(X−C))=∅ if and only if A⊂B∪C

Q: (a) Let A be a subset of R". Show that (A = (A°)°.

A: WE proved it using Claim 1 and Claim 2.

Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Prove that if R, S, and T are nonempty sets, then R×T ⊆ S×T if and only if R ⊆ S.

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Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: . Show that every map f : A → B of sets decomposes as hog where g : A → C is surjective and h : C →…

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Q: 19. If CI(A) denotes the closure of A, which of the following is not correct in general for a set A:…

A: Definition: Let X be a topological space and let A⊆X. The interior of A is the set IntA=∪V | V⊆A…

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: 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: Prove A is a subset of (A U B)

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Q: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

Q: Let X = {a, b, c}. Is the collection {X, Ø, {a}, {a, b}, {c}} a topology on X?

A: Here, the given set is X=a,b,c and collection is X,∅,a,a,b,c.

Q: Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤…

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Q: 2. Prove Lemma: For all sets A, B, and C, A x (BnC) = (A × B)n (A × C)

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Q: Let U be subset of R". Fill in the blank to define when U is an open set. U is called open if z € U,…

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Q: Let A be dense subset of (X, t)

A: Given: Let A be dense subset of x,Τ and let B be non empty open subset of X. To find: Fill in the…

Q: Exercise 60 Let p and q be two points in a topological space X. Show that there exists an open set U…

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Q: For nonempty sets A, B, and C, let f: A -> B and g: B-> C. If g is onto, then gof is onto.

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Q: Show that the subset (a, b) of R is homeomorphic with R.

A: Recall that every nonempty open interval (a,b) is homeomorphic to the nonempty interval (c,d).

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

Q: )Š is the smallest closed set in M such that S CS.

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

A: False

Q: Let {Fa}aeA be a collection of closed subsets of R. (a) Prove, from the definition of closed, that…

A: Given that Fαα∈A is the collection of closed subsets of ℝ. The objective is: To prove that ∩α∈AFαis…

Q: Prove that if A is closed and B is open, then A \ B is closed.

A: Solution:-

Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

A: Solution: Let us assume that the subset A has j elements.After fixing A, there are many choices for…

Q: B. Prove that the union of a finite number of compact subsets F1, F2, ..., F, of RP is a compact…

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Q: 6. Let S be a set with a finite number of elements, and let f: S→→ S be a map (a) If fis onto, can f…

A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: If (X,, T,), (X, T2) are two topological spaces, and A,, A, are open sets in X,and X, respectively,…

A: