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). (c, 4)} - Then the topology by f and Ty is Ty = (0, X, (a, b}}. on X induced %3D Select one: OTrue O False

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:

Q: Let T be the Euclidean topology. Let A = [0,2] and B = [0,1[, then B is clopen in (A, TA) where TA…

A:

Q: 3. Let r>0 and define the closed ball by B,(z) = {y € X : d(x,y) Sr} Show that B,(r) is closed set…

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

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A:

Q: Let (X, Tx) and (Y, Ty) be two topological spaces, such that X = {a, b,c, d}, Tx = {6, X, {a}, {a,…

A: I

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

A:

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

A:

Q: Let T be the Euclidean topology. Let A = [0,1] and B =]0.5,1] then B is clopen in (A,T) where T is…

A: YOUR ANSWER IS True.

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

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

A:

Q: 3. Find all the fixed points and classify cach of them: f(r,u) ryev,

A: Solution:- Fx,y=xye-x2-y2 Put Fx=0 and Fy=0 Fx=e-x2-y2y-2x2=0and Fy=y-2y2 e-x2-y2=0 So,  0,0,12,12,…

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: Let T cofinite topology on R and Y={5,6,7} find Ty?

A: We have to construct subspace topology with respect to the cofinite topology

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: 3. Let r>0 and define the closed ball by B,(1) = {y €X : d(r,y) Sr} Show that B,(r) is elosed set…

A: Given that X be a metric space with metric d. The given topology on X is defined as:…

Q: (a) Let Ta be the particular point topology on X = {a,b,c, d}, and f:(X,ta) → (X, Ta) is defined by…

A: In the given question given that ζa be a particular point topology on X={a,b,c,d}.so first of all we…

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A:

Q: Q.2. Let t, be the particular point topology on X (a, b, c, d), and f:(X,T)-(X, r.) is defined by…

A:

Q: {{x, z}, {x}, {y}}. Let T be the topology generated by B and Let X = {x, y, z} and B = consider A =…

A:

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: Let r, y EN such that y # 0, then y+r#0.

A:

Q: Let (X, T) topological space, where X = {a,b,c,d} and T = {X. Ø, {a}, {a,b}, {a,b,c}} Find a) Basic…

A: Given that X, T topological space, where X=a, b, c, d and T=X, ϕ, a, a, b, a, b, c.

Q: LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology.…

A:

Q: 2. Let L={x,y,z}with the topology r={L,ø,{x},{x,y},{z},{x,z}}. Verify whether L is normal or not.…

A: The given problem is related with topology. Given that L = x, y, z with the topology τ = L, ϕ, x, x,…

Q: Let r = ⟨x, y, z⟩ and r0 = ⟨1, 1, 1⟩. Describe the set of all points (x, y, z) such that r − r0 =…

A: Given: r=x,y,z and r0=1,1,1 To describe the set of all points x,y,z such that r-r0=2 Solution:…

Q: Let R be equipped with the Euclidean topology T and let Y =]10,2O[. We denote by Ty the induced…

A:

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,…

A:

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: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

A:

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

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Q: Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

A:

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

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Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

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Q: Let f: R → R be the function which satisfies for all (z. y, z) € R° that z cos(ry) and let C C R' be…

A: Here we have to find out line integral of function f(X,Y,Z).

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

A:

Q: Consider X = {1,2,3, 4, 5} where the topology is {U C X|1 € U}. Determine whether or not X is…

A:

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

A:

Q: Q5. Let (X,T) be a topological space where X = {m,n,q,r} and T = {ó, X, {q}, {r}, {q,r}, {m, q,r},…

A: Given below the answer in details

Q: Let R have its usual topology and let X = {a, b, c}. Define f : R → X by x 0 а, f(x) = с, Is X with…

A:

Q: Find all maps ¢:(Z,+) → (Q*, ×) satisfying O (a + b) = ¢(a) × ¢(b) Va, b e Z

A: We derive the function from the given properties.

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.

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A: We need to check the necessary and sufficient conditions for continuity

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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.

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.