Q. 1: If X = (a, b, c, d, e) Determine whether or not each of the following collections of subsets of X is a topology on X (1) ti = {X,0,(a},(c, d},{ a, c, d}} (2) 12 = {X,Ø, (a),{ c, d),( b, c, d}} (3) ta = {X.Ø, {a). {a, b),{ a, d),(b, e).{ a, b, c }} %3D

Q: Let X={a,b,c,d}, then one of the following is a base for some topology on X: O a. {{a,b},{a,c},{a}}…

A: Base for a Topology If X is a set, then a base for a topology on X is a collection B of subsets of X…

Q: 2. Let X = {1,2, 3, 4} and let T = {ó, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open…

A: Since you have asked multiple question , as per our guidelines we are supposed to answer only one…

Q: Let X = {1,2, 3, 4} and let T = {, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

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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: 15. If t={X,Ø,{1,4},{4,3},{1,4,a},{b}} is a topology on X-{1,2,3,4}, then: * a=4, b=2 O a=3, b=1 O…

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Q: Problem 2. Let X be a countable set and let s e X. Consider the particular point topology Tp on X,…

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Q: We define the included point topology by Tp={UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if

A: In order for a set to be dense, it is required to be equipped in Tp and also as in the set 5 is not…

Q: (a) Consider the following topology of X = {a,b, c, d, e}: T = {X, 0, {a}, {c, d}, {a, c, d}, {b, c,…

A: In topology and related branches of mathematics, a connected space is a topological space that…

Q: Let X =[0,2).Define r= {[0, a):0sa s2). 1. Show that r is a topology on X. 2. Give an example which…

A: Let X be any set and ζ be the collection of subsets of x then ζ is called as topology if it contains…

Q: Let X:= {a, b, c}, Y:= {1,2,3,4}, Ty: = {0, Y,{1,3}, {3,4}, {3}, {1,3,4}}. Let f:X → Y, f:= {(a, 3),…

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Q: 2. Let X = {a, b, c, d, e, f}. Which of the following collections of subsets of X is a topology on…

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Q: We define the included point topology by Tp={ UcR;U=ø or peU}. Let A = [3,5[, then A is dense in R…

A: In order to be dense in R, R has to be equipped in the Tp. Also as in the interval, 5 is not…

Q: Let X = {a, b, c, d, e} and7 = {X,6. {a}. {c, d}, {a, c, d}, {b, c, d, e}} be a topology on X, then…

A: Details solution given below.

Q: Let X = (1, 2, 3, 4, 5, 6) and let T1= (X, 4, {3}, (1,2,3), (2,3,4), {2}} and T2 = {X, 4, (2),…

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

Q: Q2: Let the set under consideration be N for each n EN, define u, = {n, n+1, n+2, ...} and let 1 =…

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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: 1. Let X {1,2, 3, 4}. A topology on X is a. T = {6, X, {1}, {2}} b. T = {6, X, {1}, (3, 4}} c. T3 =…

A: As this is a multiple type question, according to the Bartleby Answering Rule, only first  question…

Q: In each of the following cases compute the set, L, 3) Let A C R. of limit points of A. (i) A=[2,3] U…

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Q: Let X={a,b,c} and Y={1,2,3} and t={Y,0,{1,2}}. The function f={(a,1),(b,3),(c,2)}. Then the topology…

A: X = { a, b, c } , Y = { 1, 2, 3 } T = { Y, ∅,  {1, 2} } f = { (a, 1), (b, 3), (c, 3) }

Q: 3 let X= {ab,c},B =} {n,b}, {9c3}.5how that Whether B is a %3D %3D base for any topology on X or not

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Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: See the detailed solution below.

Q: 1. Let X = {1,2, 3, 4}. A topology on X is a. T = {0, X, {1}, {2}} b. T = {6, X, {1}, {3,4}} c. T3 =…

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Q: Let X = {1,2, 3, 4} and let T {ø, X, {1}, {3},{1,3}} be a topology on X, then {2, 4} is a. open in…

A: Our objective is to choose the correct answer

Q: Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, , {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, þ, {2},…

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Q: 13. If t={X,Ø,{2,4},{1,4},{1,2,4},{4}} is a topology on X={1,2,3,4}, let A={1,3,4}, then Int(A)= *…

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Q: 2. Let X = {1,2, 3, 4} and let T = {ó, X, {1}, {3}, {1,3}} be a topology on X, then {2,4} is %3D a.…

A: Given X = 1 , 2 , 3 , 4 and T = ∅ , X , {1} , {3} , {1 , 3} is a topology on X Then open sets of (X…

Q: 1. Let X = {1,2, 3, 4}. A topology on X is a. T = {ø, X, {1}, {2}} b. T2 = {6, X, {1}, {3, 4}} c. T3…

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Q: Let X = {a, b, c, d, e} and t = {X, Ø, {a} , {c, d} , {a, c, d} , {b, c, d, e}} be the topology on X…

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Q: Let R be equipped with the Euclidean topology T and let Y =]10,20[. We denote by Ty the induced…

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Q: 10. If t={X,Ø,{2,4),{1,4},(1,2,4},{4}} is a topology on X={1,2,3,4}, let A={2}, then the closure of…

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Q: (3) In R² with the standard topology: {(x, y) | x² + y² {(x, y) | x² + y? > 1}. = 1}, {(x, y) | x² +…

A: In a topological space R2with standard topology. A  subset O is open if at each point x in O there…

Q: 2. Consider the topology 7, = {X,ø,{a},{b,c}}_on X = {a,b,c} and 1, = {Y,ø.{u}} Y = {u,v} . the…

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Q: 3. Ir X = {a,b, c, d,e, f} and T is the discrete topology on X, which of the following statements…

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Q: Let R be equipped with the Euclidear topology T and let Y =]10,20[. We denote by Ty the induced…

A: Induced topology

Q: (b) A = (0, 1] in the finite-complement topology on R. (c) A= {a, c} in X = {a, b, c} with topology…

A: The interior of a subset S of a topological space X is the union of all subsets of S that are open…

Q: Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, p, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, ¢, {2},…

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Q: 2. Let T be the cofinite topology on R, and let A = (-x, 1) U (1, ), B = (1,2). Fine the boundary…

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Q: 3. If X = {a, b, c, d, e, f} and T is the discrete topology on X, which of the following statements…

A: True statements are: (a), (d), (g), (i), (l), (o).

Q: (2.2) Let C = {a, b, c, d} and T = {0, {a, b, c} , {b, c} , {a},C} be the topology defined on C. %3D…

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Q: 2. Let X = {1,2, 3, 4} and let T = {ø, X, {1}, {3}, {1,3}} be a topology on X, then {2, 4} is a.…

A: b. closed in (X,T)

Q: Let X = {1, 2, 3, 4, 5, 6} and let T1 = {X, %3D 4, {2}, {1,2}, {2,3,4}, {1, 2,3,4}} and T2 = {X, þ,…

A: If T is the topology on Set X then T contains all subsets of X.

Q: Let T = {X. Ø, {b}, {a, b}} be a topology on X = {a,b,c} and let A = {a,b,c}, B = {a,b,c}. Find a)…

A: Given That: Let T={X,ϕ,{b},{a,b}} be a Topology on X={a,b,c} A={a,b,c} ,B={a,b,c} To Find: a)…

Q: 4. Let X= {a,b,c} with the topology r ={X,ø,{a},{b},{c},}{a,b},{a,c},{b,c}}. Verify whether X is…

A: Given data X=a,b,c with the topology,  τ=X,ϕ,a,b,c,a,b,a,c,b,c

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 = {a, b, c, d, e} and let T ={X, Ø, {a}, {c,d}, {a,c,d), {b,c,d,e}), therm %3D O (X,T) is…

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Q: We define the included point topology by Tp={ UcR;U=Ø or peU}. Let A = [3,5[, then A is dense in R…

A:

Q: Let R be equipped with the Euclidean topology T and let Y =[10,20]. We denote by Ty the induced…

A: Induced topology

Q: Let X={a,b,c,d}   Let T1={X, emptyset, {a}, {a,c}, {b}, {b,c}, {a,b},{a,b,c}, {b,c,d}} T2={X,…

A: Let X={a,b,c,d}   Let T1={X, emptyset, {a}, {a,c}, {b}, {b,c}, {a,b},{a,b,c}, {b,c,d}} T2={X,…

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

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Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

A: