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

Q: Define a smallest possible set of which both {1,3,5} and {0, 3,4 } are subsets.

A: The subsets given are {1, 3, 5} and {0, 3, 4}.

Q: In the discrete topology on the set X, every set is open and it also follows that every set is…

A:

Q: By considering the collection of closed intervals {[1- 2+ :n € N}in R, show that the union of an…

A: Answer

Q: Prove that every infinite subset of a countable set is countable

A:

Q: define a set of which both {1,3,4} and {0,3,5} are subsets find such a set with the smallest…

A: We have to find the set with smallest possible number of elements which both {1,3,4} and {0,3,5} are…

Q: Prove that the union of countable set is countable

A:

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

A:

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: 26

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

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

A:

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

A: Countability

Q: Choose any that is a subset of the set (Ø} {{Ø),0} {{1}}

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Q: Prove that (2,5) is an open set

A:

Q: Find two infinite sets A and B so that A UB, AN B,А — В аnd B — А are all infinite sets. Show the…

A:

Q: Set is closed

A: To prove that the Cantor Set is closed.

Q: What set is in another class of infinity that is bigger than N?

A: We know that N represents set of natural number which is countably infinite. In order to get set…

Q: rsection of any collection of com

A: This is a real analysis question. Let (X,d) be a complete metric space with X not ∅. Suppose the…

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

A:

Q: Show that if A is an infinite set, then whenever B is a set, AUB is also an infinite set.

A: we have to prove that AUB is infinite set.  Since, given A is infinite set. since we know that…

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

A:

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: Let A and B be infinite sets that are uncountable. Do an and B have the same car finality as each…

A: Let A and B be infinite sets that are uncountable. Do A and B have the same cardinality as each…

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

A:

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

A:

Q: union of a collectionm of connected sets that have a point in common is conne

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Q: two example of open set in R, closed set in R, and the closure of that set

A:

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: Give two evamples that infinite intersection of open Sets not be open. may

A: See the attachment

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: 15. We have seen that an infinite set can be put into a one-to-one correspondence with a proper…

A:

Q: Define a set of which both {1, 3, 4} and {0, 3, 5} are subsets. Find such a set with the smallest…

A: {0,1,3,4,5} is the smallest set that contains {1, 3, 4} and {0, 3, 5} as sub-sets.

Q: Identify the set shown below as finite or infinite, {10, 11, 12, ..., 40} ..... OInfinite O Finite

A:

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

A: The answer is given as follows :

Q: How to Apply the Archimedes'principle to prove that a set is empty

A: Consider the given information. the Archimedes' principle is defined as, Property states that for…

Q: open

A:

Q: Prove that empty set is subset of all sets: φ ⊆ A     Please help me figure this question out.

A:

Q: Give a set that is not closed under addition

A: In the question we have to give a set that is not closed under addition.

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

A: (a) We have to prove that the intersection of an arbitrary collection of closed sets is closed. Let…

Q: Every uncountable set in a 2nd countable space has a limit point.

A: We have to prove that every uncountable set in 2nd countable space a limit point. Definition:- Let…

Q: prove that the union of a countable set and an uncountable set is uncountable

A:

Q: is the set [5,100] compact?

A: Given that:A set is S=5,100.We have to check whether a set 5,100 is compact or not?

Q: The intersetion of in finite open sets heed not to be open.

A:

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

A:

Q: Give an example of a set in R ^ 2 with the usual topology that does not have any subset (not…

A: Given that R2 with the usual topology. The aim is to find an example of the set  R2 with the usual…

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.