Prove directly that the union of two disjoint sets, where one is countably infinite and the other is finite, is countable.

Q: Select the collection of sets that forms a partitic

A: Introduction: A set of disjoint subsets of another set. The original set must be equal to the sum of…

Q: Give two sets that are equal. Prove that they are indeed equal by using the "Axiom of…

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

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

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Q: Prove a neighborhood is an open set

A: Recall that, a neighbourhood of a point is a set of points containing that point where one can move…

Q: True or False If A is a set, the complement of A is the setof all the elements in the universal set…

A: Complement of a set A: Consider a set A. Then the complement of the set A is the set of all elements…

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: Provide a counter example that infinity intersection of open set is open

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Q: 3.1 Show that a finite intersection of closed sets is closed.

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Q: 26

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

Q: Show

A: We prove this statement by using definition of Borel Set.

Q: 2. Show that any subset of a countable set is countable. 3. Show that a nonempty set S is countable…

A: Hence any subset of a countable set is countable.

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

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Q: The family of all finite subsets of a countable sets is countable.

A: Countability

Q: А - (ВnС) %3D(А — В) n (А — С)

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Q: prove that Cantor set is closed and compact.

A: Use the property that arbitrary intersection of closed set is closed and the Heine Borel Theorem.

Q: Prove that empty set is subset of all sets: φ ⊆ A

A: Given: An empty set and set A is given. To Find: To prove that empty set is subset of all sets: ϕ⊆A

Q: Is the Cantor set closed? Is it compact? Explain!

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Q: Prove: The Cartesian product of countable sets is countable.

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Q: Show that any convex open set is simply connected.

A: The objective is to prove that any convex open set is simply connected. Simply connected is nothing…

Q: Prove that every Union of Open sets is also Open.

A: To prove: Every Union of Open sets is also Open.

Q: Determine if the union/intersection of two open sets is always open then determine if the…

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Q: Prove that the Cantor set has measure zero.

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Q: uppose that ℱ is a non-empty family of sets, B is a set, and ∀A ∈ ℱ (A ⊆ B). Is ∪ℱ ⊆ B? Either…

A: Given: To explain the given statement as follows,

Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set.

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Q: There is a countable collection of compact sets whose union is not compact. Select one: O True False

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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: prove that empty set is a subset of every set

A: We have to prove that empty set is a subset of every set = ?

Q: If G(sub alpha) is a collection of closed sets, then the intersections of the G(sub alphas) is a…

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Q: Give an example of sets {An}n≥1 such that for each n ≥ 1, the set An ⊂ [0, 1] is both compact and…

A: Construct sets Ann≥1 such that for each n≥1, the set An⊂0, 1 is both compact and connected, but the…

Q: Prove that derived set of a bounded set is also bounded.

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Q: 1) Use any method(descriptive, roster method and set builder notation) to give three examples of…

A: 1) A=Set of water droplets B=Set of stars in the midnight sky. C=Set of cells in human body

Q: 5. Show that a finite union of countable sets is countable.

A: The answer is given as follows :

Q: open

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Q: Construct a family of sets that demonstrates R is not compact

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

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Q: Prove that the greatest member of a set, if exits, is the least upper bound of the set,

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Q: ) Find an example of an infinite collection of closed sets whose union is not closed.

A: Any closed interval is closed in R. From that we created this example.

Q: Prove that a non empty set A is countable if and only if there exist a bijection from A to a…

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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: Determine whether each of the following statements is true or false, and explain why.A set is a…

A: A set A is subset of another set B if all elements of the set A are elements of the set B.

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: Prove that the set of all finite subsets of ω is countable. (Set theory)

A: The objective is to prove that the set of all finite subsets of ω is countable.

Q: Prove that if A is a transitive set then UA is a transitive set.

A: If A is a transitive set then, It is proved that ∪A is a transitive set as shown below.

Q: 2. Prove that the union of any finite collection of closed sets in RP is a closed set in RP.

A: A set A is closed if and only if every limit point of A belongs to A. Now we can prove the result.…

Q: Let A1, · · · , An be countably infinite sets. Prove that A1 × A2 × · · · × An is countably…

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