Q: Choose the correct answer and attach the details of your work. 1. Let X = {1,2, 3, 4}. A topology on…
A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…
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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…
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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: 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, 5, 6} and let T1 = {X, 0, {3}, {1,2,3}, {2,3,4}, {2}} and T2 = {X, p, {2},…
A: T2 is a topology but T1 is not
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: Let X =[0,2).Define r= {[0, a):0sa s2). 1. Show that r is a topology on X. 2. Give an example which…
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Q: Let d and e be metrics on a set X such that for each ball Ba center at pE X there exists a ball Be…
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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: * Let R be with the discrete topology. If A = {1, 3,5, 7, .}, then Aº A O N O RO
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Q: Which of the following is not a base for any topology on R? a. {(a, b]: a,b ER and a <b} O b. {[a,…
A: Base for a topology: The base for topology of any set X is collection B of subsets of X which…
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: E.X; Let T = {E₁ = (-∞,a): a = R}U{R,Ø} be a topology on R and let A = [2,3), B=(3,4), C = (1,5], D…
A: It is given that, T = Ea=-∞, a : a∈ℝ∪ℝ, ∅ is a topology on ℝ. We have to find the closure of the…
Q: Let X = {a, b, c}, T1 = {X, p, {a}, {b}, {a,b}} and T2 = {X, p, {a}, {c}, {a,c}}. Then one of the…
A: Given that set X={a,b,c}
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: Consider the set A=(x:x>a, a E R}U(x:xsb, bE R). Find the topology on R which has A as subbases.
A: Given : Given a set A = x : x > a, a∈R ∪ x : x ≤ b, b∈R To…
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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: 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: 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: 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 T be the discrete topology on X. Consider the following statements: (a)…
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Q: LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology.…
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Q: Let (X, T) be a topological space, AcX, then A'next (A) = 10 AO
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Q: Which of the following is not a base for any topology on R? O {(-0,a): a E (-∞, 0]}U{(0, ∞)} O {(a,…
A: Base for a topology: Let X, τ be a topological space. A collection of open sets B is said to be base…
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: Let X = {0, p, q, r, s} and consider the topology on X below: T = {X,0, {o}, {q,r}, {o,q, r}, {p, q,…
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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.…
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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 R be with the co-finite topology. If A = {1,3,5, 7,..}, then Aº R O Q O A O N O
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Q: B) Let (R, T) be the left ray topological space If Y=(3,7], then define ty and determine the open…
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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…
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Q: Let X be an infinite set with the countable closed topology T={S subset of X; X-S is countable}.…
A: Connected means it cannot be written as union of disjoint open sets
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: 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: Q. 1: If X = (a, b, c, d, e) Determine whether or not each of the following collections of subsets…
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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: Consider R with the left ray topology. If A={2, 4, 8} the derived set of A is O a. (8, 00) Ob. A Oc.…
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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: Let X be an infinite set with the countable closed topology T={S subset of X:X-S is countable}. Then…
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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: Let X={a,b,c), then one of the fonowng is a base for some topology on X: O a. {{a,b),(b.c} O b.…
A: Use the definition of bases in topology.
Q: Let X = (1, 2, 3, 4) and T be the discrete topology on X. Consider the following statements: (a)…
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Q: Consider the standard topology on R, Let A=[3,7], the interior of A is Oa A O b.6 Oc. (3,7) OdR
A: Consider the standard Topology on R. Given A =[3,7] We need to find the interior of A.
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- 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology. Prove that thereare disjoint open subsetsVandWofXsuch thatX=V∪W.Is T ={∅, X, {1, a}, {1, b}, {2, c}, {1, a, b}, {1}}a topology on X = {1, 2, a, b, c}?Why?
- Consider X = {a1, a2, ..., an}Can a topology of X with 3 open and one with 4 open be equivalent?I, Let ¥ ={a,b,c} and B={ {a,c} ,{b,.c} } c P(X). Show thatcannot be a base for any topology r on X . 2. Let (Vr) be a topological space. Where Y ={a,6 ,¢ ,d,e } andr={X .®,{c},{d}. {ed} .{d.e} .{e.d.e}, {b,c,a}, {a,b,c,d }}Show that f° ={ {c.d},{d,e},{a,b.c}} is a subbase for thetopology +r. 3. Let X ={a,b,c,d,e}, f° ={ {a,b} , {b,c} ,{c,e},fe} } o PX)Find the topology + on X generated by /".Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7.(i) Prove that |X ∩ Y| ≥ 4. (ii) Deduce that X ∩ Y ∩ Z is non-empty. [Hint: Consider (X ∩ Y) ∪ Z.]
- If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.Let X = {1, 2, 3, 4, 5, 6}. Prove that in general, if X is finite there is only one topology for X such thatX is T1.Let M be a closed subset in (X,d). Show that b(M) is a subset of b(b(M)) where b(M) is the boundary of M.