Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

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Q: Remark. For everyr E N, a set in R" with the Euclidean metric is com- pact if it is bounded and…

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Q: Let (X, d) be a metric space and let A ⊆ X be a compact subset. show that, If B ⊆ X and d(A, B) =…

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Q: Which sets are open in the metric space (IR, d), v i). [1, 2) ii). (1, 2) U (2, 3)

A: ii). (1, 2) U (2, 3)

Q: Let (N, d) be the usual metric space, then the set A= {1,3} is O closed only O neither open nor…

A: ( N,d) be a usual metric space of natural number.  A={1,3} be a finite subset of N. Since A is…

Q: メ=D¢ c omplex number %3D Doyou CGid) is metric space of not ??

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Q: Let (x,d) be metric space and A, BC X. 2. Show that (AnB)= A'n B and AUB = AUB %3D

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Q: Let (X,t) be a topological space, A, B C X. Then A' UB' = (A U B)', A'n B' c (A N B)' Select one: O…

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If An B = A nC = BnC = 4,determine µ(AUBUC). %3D

A: Due to our guidelines we can solve only 1 question plz upload rest of questions separately.

Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

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: Q. Let (X, B(R), µ) be a Borel measure space. If AnC = ¢, BnC = ¢, µ(An B) = 0. Determine µ(AUBUC).

A: Measure theory

Q: Let (X,d) be any metric space, then the only open sets and closed sets are and X O True O False

A: Given statement is false because if we take descrete metric space

Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. (2) If ACB,…

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Q: let Qid) be a metric space and AiB Bubeet of x ACB = d(A) cd(B) [ Soi d(A) is closed set] P B is…

A: Let X,d be a metric space and A, B⊆X.   We will use the following in the proofs, Definition of clA:…

Q: A subset (a,b) ≤R is T2-space in (R,Tu) * 1 true 2 False

A: Given: A subset (a, b)⊆ℝ is T2-space in ℝ, Tu. To determine: Whether given statement is true or…

Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: ki(x, ele), ) bea be a Boral mea sure Space Suppose that ŷ An obd be afumily of lements of BiR) Show…

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Q: Given a Euclidean space E, if f: E→E is any self fanguidodge door of pas food of go

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Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

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Q: Q # 3(b) Let (G, H) be a disconnection of a topological space (X, 7), ant let AC X. If A is…

A: Before going to this proof, we prove the below statement which in turn proves the above result.…

Q: Which of the following are compact spaces? (a) S" with a finite number of points removed. (b) The…

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Q: Let A and B be any two subsets of a metric space (X, d). Then (1) A is a closed set. biso (2) If…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 3. Let K,and K, be compact subsets of a metric space X. Show K, U K2 is compact

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Q: A, B be subsets of a metric space. Show that A U B = B and that A B. Give an example to show that An…

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Q: Is the following statement True or False? Justify each answer. If S is a closed bounded subset of a…

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Q: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

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Q: Let E and F be compact sets in a metric space (X, d). Show that EUF is compact using the definition…

A: 1. Let E and F be compact subsets in a metric space X,d. Let UE and UF be arbitrary open covers of E…

Q: Let (C[a,b] .d) be the discrete metric space. Then Cla,b] is bounded. * False True

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Q: every finite subset of metric space is compact. ture or false?

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Q: A of a metric space (x, d.)

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Q: Q. Let (X, B(R), µ) be a Borel measure space. If An BnC ¢,determine µ(AUBUC).

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Q: Prove the following: Let (Y, d) be a metric space. If Y is totally bounded, then Y is bounded.

A: Given:  (Y,d) is a metric space, and Y is totally bounded To Prove Y is bounded Definition: Let…

Q: ovide an illustration of the statement "Every normal space need not be a ular space."

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Q: Every closed subset of a connected metric space is connected. O True O False

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Q: the following subsets in an indicated metric space determine the interior, boundary points and…

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Q: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Q7: Prove that ( IR", II-||) is complete metric space.

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Q: (c) If X={1, 2, 3}. T= {4, X, {1}, {1,2}, {3}, {1,3}}. JUSTIFY whether the topological space (X, t)…

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Q: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

Q: 3. Consider the metric space (R..|) and let S be the union given by (0}U UI J. Note that each of the…

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Q: Is any discrete space complete? Give an example to substantiate your answer.

A: A metric space is complete if and only if every Cauchy sequence is convergent.

Q: Every closed subset of a connected metric space is connected. True False

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Q: Remark: If (V, = « +p vi " = a + p

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Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…