Question 9. Let A=(101,102,103., n} be a subset in the usual topological space where n is an arbitrary integer. Find A'.

Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r>0…

Q: 4. (a) If X is an infinite dimensional space, show that the set {x = x | || x || = 1} is not…

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Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

Q: Question 3. Let , and r, be two topologies on X such that ,cr;. Construct a space on which a r-limit…

A: Given below the detailed solution

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: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T -…

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Q: Let ([0,1), d) is a met ric space. The set G= [0, is open in (R,d) .A O (open in ([0,1),d.B O (not…

A: Option (B) is correct.

Q: 3.2 Prove that in any metric space (S.) every closed ball Se[xo] is a closed set.

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Q: Question 5. In a topological space (X,7) a subset ACX is called regular open if A=int(cl(A)) "A= A".…

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Q: 3- A discrete topological space (X,T) is: a-To-space, b-T₁-space, c- T₂-space, d- No one.

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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: 3. Let (X, T) be the topological space defined in question 2., then the closure of {2} is a. X b.…

A: It depends on the given topological space. Here topological space is not specified. If it's usual…

Q: If X is a normed space then the open ball and closed ball is a Example: convex set

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Q: Theorem 2.28. Let A be a subset of a topological space X. Then Int(A), Bd(A), and Int(X – A) are…

A: Given that A is a subset of the topological space X. We have to prove that IntA, BdA and IntX-A are…

Q: Question 4. Let A be a subset of a topological space (X,r). Prove that the following statements are…

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Q: Question 10. Prove or disprove. Any infinite subset A of a discrete topological space (X,r) is…

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Q: Question 5. In a topological space (X,t) a subset ACX is called regular open if A=int(cl(A)) "A A".…

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Q: Problem 3. Let X be a finite set. Prove that the finite complement topology on X is the discrete…

A: Note: The finite complement topology is also called cofinite topology. And the discrete topology is…

Q: Question 9. Prove or disprove. If (X,7,) and (X,r,) are T, -spaces, then (X,7,nt,) is also a T, -…

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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: a) Consider the set X = {1,2,3} with the topology T = {Ø, X,{1}, {2, 3}, {1,2, 3}}. Show that (X, t)…

A: As per our guidelines we are supposed to answer only one asked question.Kindly repost other…

Q: Theorem 2.3. A set U is open in a topological space (X,J) if and only if for every point x€ U, there…

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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: 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: Let (N, d) be the discrete metric space, then the set A = {2,3, 4} is closed but not open open but…

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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: Question 6. Let (X,r) be a topological space and A, BCX. Determine whether (A-B)CA -B or not.

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: B) Let (R, T) be the left ray topological space If Y=(3,7], then define ty and determine the open…

A: Given: B) ℝ, τ is a left ray topological space and Y=(3, 7]. To determine: Which one of the open and…

Q: a) Consider the set X = {1,2,3} with the topology t = Show that (X, t) is a topological space. (0,X,…

A: As per our company guidelines we are supposed to answer only one asked question. Kindly repost…

Q: Question 2 Prove directly (i.e. from the definition of compactness) that if K is a com- pact subset…

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Q: Problem 5. Let X be a topological space; let A be a subset of X. Suppose that for each I E A there…

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Q: a) Consider the set X = {1,2,3} with the topology t= (0,X, (1). (2, 3}. (1, 2, 3} }. Show that (X,r)…

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Q: Question 3. all its cluster points. Prove that a subset of a metric space is closed if and only if…

A: We have to prove that a subset of a metric space is closed if and only if it contains its cluster…

Q: 1) let (X, T) be a topological space and A, BEX. In this case, show the following a) = 0 а) бала " a…

A: Given that X, τ is a topological space and A, B⊂X . To show: (a) if A is open , then ∂A∩A =ϕ…

Q: If (X, τ) is a topological space and if A,B⊂X Show that ∂(A∩B)⊂∂A∪∂B.

A: Boundary of a set in topological space: Let A be a subset of a topological space X, τ. The boundary…

Q: Task 13 Let (M, d) be a compact metric space. Show that closed subsets of M are compact.

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Q: Show that the set of all real numbers constitutes an incomplete metric space if we choose d(x, y) =…

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

Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

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Q: Let (R, d) be the metric space with d(x, y) = /[x – y[. Then the open ball O [-3, 5] ]-3, 5[ ]-1, 3[…

A:

Q: Question 2* Which one of the followings is compact a. A discrete space with at least to points. b.…

A: Given: Statements: (a) A discrete space with at least two points is compact. (b) Any set with the…

Q: Question 9. Let A (101,102,103., n} be a subset in the usual topological space where n is an…

A:

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: Question 7. Let (x,r) topological space and A, BcX. Determine be a whether A-B2A-B or not.

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Q: Question 1. Let (X,r) be a topological space and let F be the collection of all closed sets, Show…

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Q: Question 8. Let A = {10,11,12} be a subset in the usual topological space. Find A'. Question 9.…

A: Since multiple questions are given in the question, the solution to the first question is given…

Q: Let N be with the discrete topology If A=(1,3,5..), then Bd(A)=

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Q: Question 9. Prove or disprove. If (X,r,) and (X,r,) are T,-spaces, then (X,7,nt,) is also a T,-…

A: -space: A topological space is -space if it satisfies the following axiom of Frechet : "If x and y…

Q: 3.3 Let 11 and ra be distinct points in the metric space (S, d). Verify that there are open balls…

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.