Problem 17.1. Let C be a collection of subsets of the set X. Suppose that 0 and X are in C, and that finite unions and arbitrary intersections of elements of C are in C. Show that the collection T = {X -C|CinC} is a topology on X.

Q: Question 4 Show that every open subset of RP is the union of a countable collection of closed sets.

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Q: Problem 17.1. Let C be a collection of subsets of the set X. Suppose that Ø and X are in C, and that…

A: This is a problem of topology.

Q: 1. For a subset YC (X,T ) , show that the collection TY ={Y NUJU ET} is closed under finite…

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Q: An indexed family of sets (Alhet is said to be a cover of a set X if and only if X CUieIA . Use this…

A: It  has been given to us that  X⊆∪i∈I{C}i and X⊆∪j∈J{D}j. We need to show that X⊆∪(i,j)∈I×J{Ci∩Dj} .…

Q: be a collection of sets in S'. Show that we can express U E, as the disjoint union U, Er of sets in…

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Q: 2-2. Let X = {1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic…

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Q: 6. A relation R on a set X is called a total order if it is antisymmetric, reflexive, transi- tive,…

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Q: (1) Let X be a set and tEX, let t={GCX/ t€G or G° finite }, show that whether t is a topology on X…

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Q: 10. Prove that for subsets X, Y, and Z of a given universal set U (a) X-Y=X-(XnY)= (XUY) - Y (b)…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

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: Exhibit a countable collection of open sets U; such that each open set OCR can be written as a union…

A: Given :- Uj is countable collections of open sets. To prove :Every open set O⊆ℝ can be written as a…

Q: Show that there is no topology on X={a,b,c,d,e} based on the family B={{a,b},{a,b,d},{b,d,e}}

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Q: Problem 8. Let Rf be the set of all real numbers with finite complement topology. In this topology G…

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Q: Theorem 2.47. Suppose B is a basis for a topology on X and Y C X. Then By := {BnY| BE B} is a basis…

A: Definition: Basis for a topology Let X, τ be a topological space and Y be a subset of X. A…

Q: 5. If {T,} is a family of topologies on X, show that Ta is a topology on X. Is UT, a topology on X?…

A: First we will prove the result that arbitrary intersection of a topologies is a topology by showing…

Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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Q: (A,7) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit points…

A: Solution: (4) Answer: The set is  A=1,2,3 Clearly, set A is finite.  We know that a finite set does…

Q: 6. Let's say the set X= {1, 2, 3, 4 is given as the whole space and M = {Ø,{3},{4},{1, 2},{3, 4},…

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Q: 4) LetXbe any set with the discrete topology. Let Y be any subset of X. Prove that ô( Y ) =Ø.

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Q: 5. Give an example of a collection (A,, of sets with the properties that: (a) A, # Ø for each i> 1,…

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Q: 3. Let X be a topological space. Show that for every subset A of X and for ever U we have AQU =…

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Q: 5. Let (X, d) be a metric space such that the intersection of open set and closed set is always…

A: As per the question we are given a metric space (X, d) space such that the intersection of open set…

Q: Problem 3. Let X be a finite set. Prove that the finite complement topology on X is the discrete…

A: Note: The finite complement topology is also called cofinite topology. And the discrete topology is…

Q: 7. Give an example of an indexed collection {An nE N} such that 1 An and U An = Aj = N.

A: Given that An|n∈ℕ To give an example of an indexed collection  An|n∈ℕ such that ∩n=1∞An=∅ and…

Q: (a) When is a subset of a topological space said to be closed? (b) Fotmulate the definition of…

A: If X,𝞽 is a topological space, then a closed set in X is a subset of X whose complement is a member…

Q: shows that there is no topology in X based on the family B = {{a, b}. {a, b, d}, {b, d, c}}

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Q: Question 5. Let S be a subbase for a topology r on X and let A be a subset of X. Show that the class…

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

Q: Question 1. Let X and Y be infinite sets. Let X be equipped with the discrete topology, and Y…

A: Given, X and Y be two infinite sets in topological space.  X is a discrete topology and Y is a…

Q: 3) Let D = {(x, y) = R² | 0 < x² + y² ≤ 4}. Give all boundary points of D. Is D open, closed or…

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Q: 7. Show that the Obenchain class LOb(B) is a convex set.

A: Given: To show the Obenchain class LOb (β) is a convex set is given as,

Q: 4. For the given collection of sets {[-1,3). [-1,2), [-1,23). [-1,2),...} define a set A, for each…

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Q: Problem 5. Let X be a topological space; let A be a subset of X. Suppose that for each I E A there…

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Q: メナy= 3.28 %3D X -Y = 2.58 X -1= 0.78

A: Given equation is, X+Y=3.28.....1X-Y=2.58.....2X-Y=0.78.....3

Q: 3. Let X be a set. Call a subset A C X to be finitely complemental if its complement is either…

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Q: 4. A space X is called separable if X has a countable dense subset. Which of the following is not…

A: see below

Q: Let (X. T) be a T, topological space and A a nonempty subset of X, if x is a cluster point of A…

A: Given: Let (X,T) be a T1 topological space and A a non-empty subset of X, if x is a cluster point of…