Let X = {a, b, c}, T1 = {X, 0, {a}, {b}, {a,b}} and %3D T2 = {X, p, {a}, {c}, {a,c}}. Then one of the %3D following is true: the smallest topology on X containing both T1 and T2 is the discrete topology. O T1UT2 is a topology on X but T10T2 is not O TINT2 and T1UT2 are both topologies on X T1NT2 and T1UT2 both cannot be topologies on X

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: 29. Let N be with the co-finite topology If A={1,3,5.), then Bd(A)=

A:

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 X = {a, b, c, d, e, f, g} and B = {{a,c, e, f}, {b, c, d, f,g}, {c}, {f}}. Show that B is a base…

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Q: 1. Let X = {a,b,c} and B={{a,c}.{b,c}}cP(X). Show that B cannot be a base for any topology r on X.

A: According to the answering guidelines, we shall solve first question only. If you want others to be…

Q: and C) Define the neighborhood system. If X=(e₁,e₂,ez.e. es. e) T= {p,X, (e), (e₂). (e₁,e₂), (e₁,e).…

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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: (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 Z be with the discrete topology.11 If A = {1,3,5, 7,...}, then Aº %3D %3D A O N O R O

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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: option (d) is correct.

Q: In a Cofinite topology of X the collection of subsets a) r = {AcX|A ±Ø or A is finite} b) r = {Ac X|…

A: Let X be a non-empty set, then the collection of subsets of X whose complements are finite along…

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

A: Explanation of the answer is as follows

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: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

Q: 7. Let X = {a,b,c}. Show that Bß ={{a,b},{b,c}}cannot be a base for any topology on Х.

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Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

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Q: Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed…

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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: The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product…

A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: Let T={X, {a}, {a,b).{a,c,d}, {a,b,c,d}, {a,b,c}} be a topology on X=(a,b,c,d,e). (ii) Find N.…

A: Given : (X, T) be a topological space,  where  X = {a, b, c, d, e} and T = {X, ϕ, {a}, {a, b}, {a,…

Q: 1. Let X = {1,2, 3, 4}. A topology on X is %3D a. T = {6, X, {1}, {2}} b. T2 = {0, X, {1}, {3, 4}}…

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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 &gt; a, a∈R ∪ x : x ≤ b, b∈R     To…

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: Consider the following subsets of R equipped with the Euclidean topology: A = { neN} Зп — 1 n+1 ° B…

A:

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 =…

A:

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

A: Given that X is an infinite set with the countable closed topology T-S subset of X; X-S is…

Q: Let X = {1, 2, 3, 4} and T be the discrete topology on X. Consider the following statements: (a)…

A:

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: Q2) True or false (1) B = {(a), (b), {c}} is a base of the discrete topology on X = (a, b, c).…

A: Let X be any set, then collection of all singletons is basis for discrete topology on X.  B = {{x} :…

Q: Let S be an open subset in C and f EC¹(2). Let R be a closed rectangle with vertices v₁ = a +ic,…

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Q: If X = {a,b, c}, let T1 = {0, X, {a}, {a,b}} and T2 = {0, X, {a}, {b, c}} . Find the smallest…

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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: 18: Consider the two linear maps T, and T2 on V, defined as T(X, X, x) = (0, xz x) and T,(x , x, 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…

A:

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: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

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…

A: Given that X is  an infinite set with the countable closed topology T =(S subset of X: X-S is…

Q: A topological space (X, t) is a T1-space if and only il for each x e X, the singleton set {x} is…

A:

Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

A:

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

A: here option (c) is true because

Q: 1. Let X be any nonempty set and let p E X, Prove that : E, {G X:p¢ G}U{X} is a topology on X.

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

Q: the collection of all close subsets of R is called the topology of R ylgn İhi

A: False