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 S, = {Ans:SeS} is a subbase for the relative topology r, on A.

Q: Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable).…

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Q: Consider the collection of sets {An: n E R}, where for each n E R An := {(x, y) E R²: 2x + n}. y =…

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Q: Let X:= (a, b.c), n=(1,2,3.4), Ty:= (0,Y, {1,3), [3,4}, {3), [1,3,4}}. Let fx-Y. f:= {(a,3). (b, 3).…

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Q: Let S be set of all strings of a's and b's. Define a relation R on S as follows: Vs, t E S, sRt e…

A: Let A be any and let R be any binary relation on the set A. Relation R is symmetric if for a, b∈R,…

Q: 1 0 10 1 0 1 0 1 0 Let A = {1,2,3,4,5} and M, = 1 1 1 1 1 0 1 0 1 0 |1 0 1 0 1 a) Give the relation…

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Q: Let T be the finite closed topology on a finite set X, then * O Tis the Euclidean topology O Tis the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (b) Let A be a partition of X. Prove that R = {(x, y) = X : there exists i such that x € A; and y €…

A: A is a partition of X.R=x, y∈X: there exists i such that x∈Ai and y∈Ai

Q: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection ?…

A: Let A be a nonempty subset of B. Is it true that the inclusion mapping from A to B is an injection?…

Q: Is it possible for a relation R on a set X to be both symmetric and antisymmetric? If so, give an…

A: Let R be a relation on set A .then SYMMETRIC relation: ANTISYMMETRIC relation: Recall: Identity…

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: Let Z be the set of all integers and let R be equipped with euclidean topology t prove that tr the…

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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: Consider the set A=(x:x>a, a ER)U(x:xsb, bER). Find the topology on R which has A as subbases.

A: given a set  A=x:x>a,a∈R∪x:x≤b,b∈R to find the topology on R which has A  as subbases

Q: Define a relation on Z by aRb if and only if a divides b. Is R an equivalnce relation? Why or why…

A: To define a relation on the set of integers Z, denoted by "aRb," where "a" divides "b," we need to…

Q: Q1.Consider X ={a,b, c, d, f, g} .Let T={ó, X, {a,b, c} ,{g};{a,b, c, g}} (a) Prove that T is a…

A: Since you post a question with multy part, we solve only first three, if you need any particular…

Q: The quotient topology actually defines a topology.

A: The quotient topology  actually defines a topology

Q: Prove that closed balls are closed sets in the standard topology on R?.

A: Topology Question

Q: Which one of the following statements is true?

A: In the R with the Euclidean topology and R with the finite closed topology, f might not be…

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: Show that the collection of all sets which are union of sets of the form [a, b) is a topology on R.

A: To Show: The collection of all sets which are the union of sets of the form [a, b) is a topology on…

Q: QUESTION 3 If A = {2, 3, 4, 5} and R is a relation on A such that x R y if R = {(x, y) : (x + y) is…

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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: f is said to be an open mapping iff the image by f of any open set is open. Let TI be the lower…

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Q: Show that if a topological space has a finite number of points each of which is closed then it has…

A: Topology question.

Q: Let R be an equilene relation defineed an IN as follows ! 2-8(x)ENIN Y-y alivisible by3}

A: It is given that let r be an equivalence relation defined an 1n as follows R={(x,y) belongs to N*1N…

Q: Let X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and %3D T2 = {X, p, {a}, {c}, {a,c}}. Then one of…

A: let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:   a) Both…

Q: 1) let R be a set of real number and T={Ø, R, A,, aER), where A=(a,0). Show that whether T is a…

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Q: Let C = {a, b, c, d}. Find a relation R on C that has exactly 3 ordered pair members and is both…

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Q: Prove the following: Let X denote the set {0,1} with the discrete topology. Let Y = X" = || X; ieZ+…

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Q: Consider the subset R = R × R − {(x, x)|x ∈ R}. What familiar relation on R is this? Explain.

A: Concept used: Subset relation Subset relations are used to define partial orderings on sets. A set's…

Q: Let τ and τ´ be two topologies of a set X. Prove that the family τ∩τ´ formed by the open ones common…

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Q: Let X = {a, b, c, d} and T = { X, p, {a,b}, {c,d}}, then * An open proper subset of X may be clopen…

A: To find the true options for the given questions. Definitions: 1. Open sets can be defined as the…

Q: 5. Let X and Y be sets and let f: X→Y be onto. For all be Y, let As = f-¹({b}). (a) Prove that P =…

A: Given that X and Y are the sets , f : X→Y be onto function and ∀b∈Y, Ab=f-1b. To prove: (a) P is a…