Question 1. Let X and Y be infinite sets. Let X be equipped with the discrete topology, and Y equipped with the trivial topology. Describe the open sets of the product topology on X × Y.

Q: If  X is a countable set show that every topology on X is second countable.

A: To Show: If  X is a countable set show that every topology on X is second countable.

Q: LAUBUC = ANĒNT. %3D

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Q: Question 17 How many elements are there in the relation S on the set {1, 2, 3,..., 100}, where S =…

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Q: Question 7. Let (X,r) be a topological space and A, BCX, Determine whether A-B2A-B or not.

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Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

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

Q: Theorem 2.8: The closed subsets of R have the following properties: (a) R and Ø are closed sets. (b)…

A: Note : A set is said to be closed if it's complement is open. Theorem used: Arbitrary union of open…

Q: Q3) Discuss these examples (a) Let X be an infinite set and t is the cofinite topology on X. Then…

A: We shall solve these problems in next step. Please read the answer carefully and slowly to…

Q: A closed set in the finite complement topology on R.

A: Def: A set C is closed in topological space, if complement of C is open set.

Q: I? B is a family of subsets of X Such that: (2) If B, ,BzE/ß them B,nB, EB Then B is a base for a…

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Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

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: Prove that T is the discrete topology for X iff every point in X is an open set

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Q: Theorem 3.16. Suppose X is a set and S is a collection of subsets of X. Then S is a subbasis for…

A: Given that X is a set and S is a collection of subsets of X. Let B be the family of finite…

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

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Q: For any infinite set X, the co-countable topology on X is defined to consist of all U in X so that…

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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: 1. Determine if the given collection of subset of X={a, b, c} is a topology on X. a. T1 = {Ø, X,…

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Q: If X #0, compare the following topologies on X: a) The co-finite topology 7, and the co-countable…

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Q: Which one of the following statements is true? * O Rwith the Euclidean topology and R with the…

A: Which of the following statement is true?

Q: Question 1. Let (X,7) be a topological space and let F be the collection of all closed sets, Show…

A: Closed Sets: A subset of a topological space is said to be closed if the set is open De Morgan's…

Q: Which one of the following statements is true? O R with the Euclidean topology and R with the…

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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: Which one of the following statements is true? * R with the Euclidean topology and R with the…

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Q: Question 1. Let A be a subset of a topological space (X,7) with the property that each point pEA…

A: Given below the detailed solution.

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: Question 9. Let A=(101,102,103., n} be a subset in the usual topological space where n is an…

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Q: Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology…

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Q: Define the Cartesian product of sets X and Y? How is this Cartesian product denoted?

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Q: For sets A and B, define A+B = {a + b: a € A, b E B}. (a) Prove that if A and B are open, then A+ B…

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Q: Consider the number line R with the topology t = {Ø, R, (-0, a); a E R}. Is the set A = (-3, 0]…

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Q: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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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…

Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

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Q: Which one of the following statements is true? * R with the Euclidean topology and R with O the…

A: Any two homeomorphic spaces always shares the same Topological property. This means if one of them…

Q: Question 7 Let A be a subset of R'. Let A denote the intersection of all closed sets containing A.…

A: We will solve all the three parts for you.

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is in T, then T is…

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Q: Which one of the following statements is true? * Rwith the Euclidean topology and R with the finite…

A: Answer

Q: Let X be an infinite set and I be a topology on X. If every infinite subset of X is in T, then Tis *…

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Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

Q: R is not connected if T is * the indiscrete topology the trivial (usual) topology the finite closed…

A: Option 3 is correct.

Q: Question 3. Consider the usual topology U on the set of real numbers. Describe the relative topology…

A: Since, this is a multiple question I am solving only first question for you. By (barleby policy)

Q: Question 9. Let A (101,102,103., n} be a subset in the usual topological space where n is an…

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Q: Question 7. Let (x,r) topological space and A, BcX. Determine be a whether A-B2A-B or not.

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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: Question 1. Let (X,r) be a topological space and let F be the collection of all closed sets, Show…

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Q: Which one of the following statements is true?* R with the Euclidean topology and R with the…

A: Which one of the following statement is true

Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…

A: The complete solutions is in given below

Q: If K is a compact subset of RP and z is a point of RP, then the set K, = {r+y :y € K} is also…

A: Given that K is a compact subset of ℝp and x is a point of ℝp. The objective is to show that…