Theorem 3.2.14. (i) The union of a finite collection of closed sets is closed.

Q: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: Theorem 3.7: A subset A ofa metric space (X, d) is closed if and only if /.contains all its limit…

A: We have to prove that A is Closed if and only if A contains all its limit points. Note : In proof i…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: 1.4.10. Prove that the closed interval [0, 1] and the open interval (0,1) have the same cardinality…

A: (Schr¨oder-Bernstein theorem) :Let A and B be sets. Suppose there are injective functions f : A → B…

Q: 3.1 Show that an infinite intersection of closed sets E k = 1,2,3,., in a metric space (SA) is a…

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Q: 4. Let (X, T) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit…

A: Given problem is :

Q: Every closed and bounded subset of X is compa The subset {z EX: |||| ≤ 1} of X is compact... X is…

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Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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Q: Theorem 6.20 (Heine-Borel Theorem). Let A be a subset of R" with the standard topol- ogy. Then A is…

A: Given :- Let A be a subset of ℝn with the standard topology. To Proof :- A is compact if and only if…

Q: Theorem 3.7: A subset A of a metric space (X, d) is closed if and only if/.contains all its limit…

A: A subset A of a metric space (X, d) is closed iff A contains all its limit points.

Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2,3, . (S, d) is a closed set. in a…

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Q: 3.1 Show that an infinite intersection of closed sets F, k = 1,2, 3, ..., in a metric (S, d) is a…

A: Let {Fk}  be an arbitrary collection of closed sets . . 1  i.e. for k = 1, 2, 3, . . . . . . . . To…

Q: Theorem 8.6. Let C be a connected subset of the topological space X. If D is a subset of X such that…

A: Let C be a connected subset of the topological space X.Also given that D is a subset of X such that…

Q: Theorem 6.2. Let C be a compact subset of Rstd- Then C has a maximum point, that is, there is a…

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Q: Theorem 2.1. Let{U}{_, be a finite collection of open sets in a topological space (X,T). Then N, U…

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Q: Theorem 6.3. If X is a compact space, then every infinite subset of X has a limit point.

A: We have given that, X is Compact space.

Q: Theorem 3.5. Finite Intersection Property If {Ka} is a collection of compact subsets of a metric…

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Q: (4.2) Let X = {1,2, 3, 4} and 7 = {0, {1,2,3} , {2, 3},{1},X} be the topology defined on X. (a) Show…

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Q: 2.5 For each of the following properties, give an example consisting of two subsets X, Y CR', both…

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Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

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Q: Hausdorff space.

A: Given: A set X with co-finite topology and  Hausdorff space. To Find: Under what condition a set X…

Q: Let E₁ and E₂ be open dense subsets of a topological space X. (a) Show that E₁ E2 is dense in X. (b)…

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Q: Question 2. space is the union of a finite number of closed balls radius e. Prove that a metric…

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Q: 2.6.13 LetE be a subset of a metric space X. Prove that E has empty interior (E° = Ø) if and only if…

A: Dense set and interior set

Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T, -spaces, then (X,7,nt,) is also a T, -…

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Q: Theorem 2.25. Suppose that (X, p) is a metric space. (a) The intersection of any collection of…

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Q: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Show that any infinite subset A of a discrete topological space X is not compact.

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Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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Q: 3.1 Show that an infinite intersection of closed sets Fa, k = 1, 2,3,.., in a metric space (S, d) is…

A: The given is related to Metric spaces. The solution is explained below.

Q: in the discrete metric space: 1. All collections are clo

A: Let (X, d) be a metric space. Consider a closed ball Brx0 with both center x0 ∈ X and radius r &gt;…

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

Q: 1. If X={1,2,3,4) and T={{1,2), (3,4)} then T is 2. The collection T of all rectangular regions…

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Q: (a) Prove that every nonempty proper subset of a connected space has a nonempty boundary. Is the…

A: As per Bartleby guidelines, for more than one question asked, only first is to be answered. Please…

Q: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

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Q: Theorem 2.15. Let (X,T) be a topological space, and let U be an open set and A a closed subset of X.…

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Q: 5. Prove the following theorem from class: Theorem 1. Suppose that X and Y are nonempty finite sets…

A: In mathematics, a surjective function (also known as surjection, or onto function) is a function f…

Q: Theorem 2.31. In the standard topology on R", if p is a limit point of a set A, then there is a…

A: Considering p as a limit point of set A in the standard topology on  ℝn. Here Kelley's Lemmas…

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: Theorem 6.5. A space X is compact if and only if every collection of closed sets with the finite…

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Q: 4. Let (X,T) be the topological space defined in question 2. Let A = {1,2,3}, then the set of limit…

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Q: Let X be ametric space. If A and B are each dense open subsets of X then AOB is dense open subset of…

A: First we have shown the intersection is open. Then we use the property that if a set intersect every…

Q: 2. Let (X, 1) be any topological space and (X, Tf) be the finite complement topological space. Show…

A: Let (X, τ1) be any topological space and (X, τf ) be the finite complement topological space.Show…

Q: 3.1 Show that an infinite intersection of closed sets F, k 1,2, 3,..., in a metric snace (S, d) is a…

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Q: Example 2.17. On N, let {{0,1}, {2,3}, {4, 5},..} be a subset of a topology Tad. What other sets…

A: Suppose there exist a set S which has total number of subset equal to T. The collection of all…

Q: Question 9. Prove or disprove. If (X,r,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T,-…

A: -space: A topological space is -space if it satisfies the following axiom of Frechet : "If x and y…

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: Theorem 3.60. Let X be a Hausdorff space. If x and y are sequential limit points of X with respect…

A: Theorem 3.60: Let X be a Hausdorff space. If x and y are sequential limit points of X with respect…

Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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