Suppose that R (the set of real numbers) is equipped by TO (the Euclidean topology), T1 (the discrete topology) and T2 (the finite closed topology). Then ]a,b] is: open is TO and T1 but not open in T2 open in TO, T1 and T2 open in T1 but not open in TO and T2 not open in any of T0, T1 and T2 O O O

Q: Let X be any infinite set with the co-finite topology. Identify which of the following separation…

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Q: Theorem 2.8: The closed subsets of R have the following properties: (a) R and Ø are closed sets. (b)…

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Q: Prove that the set {a E R: a2 not equal to 6} is open.

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Q: 4. Let A and B be two closed subsets of the connected topological space X. Suppose that X = AUB And…

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

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Q: Q8) Consider the set of real numbers R. Define F as follows: For a subset U of R, F = {U CR: R\U is…

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Q: 3. Prove that the sets A = {2.n+1:n € N} and B = {3⋅n+2:n € N} of the odd numbers and the "multiples…

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Q: 2. Provide an algebraic proof (use set identities and definitions only). Let A, B e N. Prove that…

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Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: A and B are bounded subsets of R, then prove that UB and AnB are bounded.

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Q: Let R (the set of real numbers) be equipped with the Euclidean topology, then A, a finite subset of…

A: Every finite set in Euclidean topological space is closed

Q: Let there be set of a, b, c. Prove A×(B∩C) = (A∩B)×(B∩C)

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Q: Given the following relation on sets : R={(x,y)1x is a proper subset of y}. Which properties hold…

A: The given relation:R=x,y| x is a proper subset of y.

Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

Q: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: 3) Let X be a set with > 3 elements. Show that there exist bijective functions f :X → X and g : X +…

A: Let x=1,2,3 then, f=123132g=123321 `

Q: Let A and B be finite sets such that B = {b1, b2}. Prove that |A – B| < |A|.

A: We are given that A and B are two finite sets, where B= {b1, b2}.We have, |B| = 2, and let |A| =…

Q: Is the set {(a₁, A2, A3, ...) € l² : |an| ≤ compact in 2? √n

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Q: What is the interior of S? b. What is the boundary of S? c. Is S open, closed, or neither? All of…

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Q: Let a, b, c. and d be any real numbers such that a < b and c < d. Prove that [a, b] is equivalent to…

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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: Find sets A, B and C such that A ∩ B,    B ∩ C and A ∩ C are non-emptysets and A ∩ B ∩ C = φ.

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Q: Suppose F and G are families of sets. Prove that (UF)n (UG) ≤ U(F N G) iff VA EFVBE G(An BCU (FNG)).

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Q: 11. Show that the set of real numbers with the usual topology is not homeomorphic to the R set of…

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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: Suppose that R (the set of real numbers) is equipped by TO (the Euclidean topology), T1 (the…

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

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Q: Suppose E is a bounded, nonvoid set of real numbers. Define |E| to be the set { x: x = E}. Is it…

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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: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

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Q: show that empty set, X is an element of S intersection T in topology

A: Here We have to show that empty set must be in S∩T where S and T are topology on set X.

Q: [4] Let A andB be sets such that A C B. Prove that A OBC B, using an argument that starts with "Take…

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Q: Given that S,T CR are nonempty, bounded subsets of the set of real numbers. State whether True or…

A: Given that S, T⊂ℝ are nonempty, bounded subsets of the set of real numbers. Then supremum and…

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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