Give an example of sets {An}n≥1 such that for each n ≥ 1, the set An ⊂ [0, 1] is both compact and connected, but the union of An n ≥ 1 is neither compact nor connected.

Q: (a) The intersection of any collection of closed sets in X is closed (in X). (b) The union of any…

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Q: Find an example of a nonempty set satisfying: • Open, every point is a limit point • Closed, every…

A: Just because a nonempty open set contains only limit points does not mean that the set contains all…

Q: Prove that the union of countable set is countable

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Q: Which of the following is true: Every bounded subset of (R, Tusuat) is compact. Every closed subset…

A: We have to find correct statements out of given statements.

Q: 8. Disprove that every compact set has only a finite number of isolated points. V Note that the…

A: Let C be the compact set.If possible suppose that 'C' has a infinitely many isolated points. say…

Q: Does there exist a non-empty open set A and two elements in A that are farthest apart (among all…

A: There are no two elements that are farthest apart in a non-empty open set. Justification : Let A…

Q: 2- Every finite set in any topological space is a- Regular set, b- connected set, c- Hausdroff' set,…

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Q: Provide a direct proof that a sequentially compact set must be both closed and bounded without…

A: WE GIVE A DIRECT PROOF. First, we prove that set A is closed then A is bounded.

Q: (ii). Prove that a set S CR is everywhere dense iff it intersects every non-empty open sets in…

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Q: Prove that intersection of any class of convex subsets of N is convex?

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Q: Closed, every point is a limit point, nowhere dense in R

A: We have to find a subset in R such that it closed ,every point is limit point and no where dense in…

Q: Using contradiction method, prove that for every nonempty sets A and B, An (B- A)=0.

A: The given problem is to prove the following set identity using the contradiction method.

Q: Show in the discrete metric space: 1. All sets are open (especially single points); 2. All…

A: We first define Discrete space

Q: 26

A: The objective is to find an infinite collection of closed sets whose union is not closed.

Q: Find an example of a nonempty set satisfying: 1. Closed, Dense in R 2. Open, Dense in R ~1. & 2.…

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Q: Show that any finite union of Nowhere Dense Sets is Nowhere Dense.

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Q: Show in the discrete metric space: 1. Only finite sets are compact: that is, if k is compressed then…

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Q: A topological space is dis connected if and only if it contain non empty set which is both open and…

A: I give you two simple example with explanation

Q: Which of the following is true? O Finite intersections of compact sets are compact, but there exists…

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Q: Using the fact that "a countable union of countable sets is countable", show that the statement that…

A: Please see the below picture for detailed solution.

Q: 2- consider the set of all finite subsets of N, is this set countable or not, prove it?

A: Consider the set of all finite subsets of N, is this set countable or not, to prove it.

Q: 8. Give an example to show that the arbitrary intersection of open sets is not an open set. 9. Give…

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Q: (ii). Prove that a set SCRis everywhere dense iff it intersects every non-empty open sets in R.Prove…

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Q: A and the set B = {x, y}, then A X B is denumerable as well.

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Q: By considering the collection of intervals {(-) : n € N} show that the intersection of an infinite…

A: Let  Sn=-1n,1n :n∈N      where N is set of natural number. for n=1,2,3,...... S1=-1,1…

Q: Give an example of a bounded set that has infinitely many isolated points

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Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

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Q: subset

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Q: Prove that the unoin of a finite number of bounded sets is bounded.

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Q: Tet X and Y be infinite sets prove that XUY is infinite

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Q: Show that the lower limit topology on R is strictly finer than the standard topology on R. That is:…

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Q: Give an example of a countable collection of bounded sets whose union bounded, and an example where…

A: Let A1 and A2  are bounded sets

Q: (i) Show that a non-empty subset of real numbers A is nowhere-dense if and only if the interior Aº…

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Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

A: Schroder-Bernstein Theorem: If there exists injective functions f : A→B and g : B→A between the sets…

Q: Verify that the two definitions of set 'A is nowhere dense' are equivalent: 1. A set is nowhere…

A: We will prove that (1) implies (2) and (2) implies (1). Thereby the two definitions will be…

Q: If both sets A and B are enumerable then its intersection is at most enumerable. If the sets A and B…

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Q: 2. For any n E N, let G, denote the interval (-). Prove that each G, is an open set in R and each G,…

A: Here Gn=-1n,1n     so, G1=-1,1           G2=(-12,12)           ..           Gn=-1n,1n Since all Gn…

Q: The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as…

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Q: Prove that the smallest member of a set, if exits, is the greatest lower bound of the set. Correct…

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Q: Explain the idea of cardinality. When is a set denumerable? Show that the union of two denumerable…

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Q: Prove that a set S⊂R is everywhere dense iff it intersects every non-empty open sets in R. Prove…

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Q: show vacuous proof of statement that "empty set is a subset of any set"

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Q: 3. Let A and B be effectively enumerable sets of natural numbers. Show that the sets AnB and AUB are…

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Q: Show wheatear an open interval (m, n) is a union of a countable collection of closed sets or not.

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Q: how that the union of a finite. number of bounded sets is bounded

A: A statement is given as, 'the union of a finite number of bounded sets is bounded. The objective is…

Q: Please solve part (a).

A: Hi! We are answering only the first two problems. If you need the solution of rest questions, then…

Q: On a set X, consider the collection consisting of four of its subsets, given by Γ = {X, ∅, A, B},…

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