An indexed family of sets (A,hcs is said to be a cover of a set X if and only if X CU,crA Use this idea to prove that if (Ches and (D,), are two different covers of a nonempty set A then (Gn DlneN forms a cover of A

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: 5. Let CI(A) means the closure of A. A subset A of a space X is called preopen if A C Int(CI(A)).…

A: Given ClA is the closure of the set A and a subset Aof space X is said to be preopen if A⊂IntCl(A).

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A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

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Q: (b) Show that the continuous image of compact set is compact but the convers need not to be true.…

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Q: For any collection a of subsets of X, there i ntaining a.

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Q: or which compact sets A does there exist an m e N such that every open cover of A contains a bcover…

A: Please check the answer in next step

Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U ·. U Bn equal A exactly?…

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Q: Let A be a denumer able set. Prove that A has a denumer able subset B such that A - B is…

A: Let A =set of multiples of 4, B=set of multiples of 3. Both are denumerable. Intersection of A and B…

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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: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let (x, eA) be a measurable space, £: X (-∞, w) and 9:* (-∞0, ∞o) are measurable. Prove that the…

A: Given that X,A is a measurable space. The objective is to prove that the sets  {x∈X:f(x)<g(x)},…

Q: rcise: Prove that every finite set S of R" has no accumulation point.

A: Proof of the the given theorem is given as

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is…

A: We prove this via contradiction. We assume that f(X) is not connected then X is not connected. And…

Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

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Q: If B1,..., Bn is a finite open cover of a compact set A, can the union B1 U... U Bn equal A exactly?…

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Q: Show that the subset A of cantor set is not homeomorphic to A+A and A*A

A: Since, A is a subset of counter set ∩nFn and A1=A+A is also a subset of ∩nFn then, A2=A*A⊆∩nFn Case…

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Q: Any connected component of Xis a closed set and X is disjoint Union of ts connected components,

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Q: Prove that a closed convex set in R" has a finite number of extreme points if and only it is the…

A: A closed convex set in ℝn has a finite number of extreme points if and only if it is the…

Q: Prove that the Cantor set has measure zero.

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

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Q: Question No 1 (8). Prove that the discrete topology on an uncountable set does not Satisty the…

A: Note- Let X be a topological space, then X satisfies the second axiom of countability or is second…

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Q: (a) If {Ta} is a family of topologies on X, show that N Ta is a topology on X. Is UTa a topology on…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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: Find an example of on a set X (G has or H of G which has le

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Q: Every proper simply connected oper equivalent to an open subset of the

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Q: Show that X is connected if and only if the only clopen subsets are Ø and X.

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Q: ) We know that {-5,-4,-3,-2,-1} and N are wellordered sets with respect to the trivial ordering. Let…

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Q: Let (X,τ) is a topological space and A ⊆ X. If all subsets of A are closed in X, then set A cannot…

A: Given that X,τ is a topological space. Let's prove this theorem by contradiction. So  X,τ is a…

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Q: Let S be a set of 100 points in the plane and the distance between any two points of S is at least…

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Q: An open proper subset of X cannot be clopen An open proper subset of X may be clopen and may be not.…

A: Solution: Definition of Clopen set: Clopen set: A set which is both open and closed is called a…

Q: Theorem 2.20. For any set A in a topological space X, the closure of A equals the inter- section of…

A: Let A  be a set in the topological space X. To prove that the closure of A  is the intersection of…

Q: One of the theorems is called well-ordered theorm: .a For any set A ,we have |A|<|P(A)| .b If S is…

A: The well-ordering theorem, also known as Zermelo's theorem, states that every set can be…

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Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: An indexed family of sets (Aher is said to be a cover of a set X if and only it xSUieIA Use this…

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Q: Prove that the union of any number of open sets in Ris open.

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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: For any collection a of subsets of X, there is ntaining a.

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Q: (x,7) is a topological space, IcCo,1] and {A;liez be Fami ly of subsets of x Uier Ai = JieI A: is…

A: Note that,  Ui∈I  Ai  C— Ui∈I         Ai   (: Ai CAi  for all i∈I) By the condition  Ui∈i Ai is a…

Q: 1.82 Let E CR be any unbounded set. Find an open cover of E that has no finite subcover. Prove that…

A: We will proceed as follows