Let (X, t) be a topological space and A, B EX, then 1- (X- A)° CX-Ā 2- (X-A)° 2X-Ā 3- (X- A)° = X-A 4- (X- A)° = (X A)

Q: 1- a = (1, .,0) in the usual metric space R. 2- (0,1) in the usual metric space R. 3- {(x, y): xy =…

A: As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly repost…

Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

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Q: The graph of f(x,y)=y2-x2 on XYZ space has a saddle point (a,b) if... Choices: A) D(a,b)=0 B)…

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Q: 2. An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two…

A: We know that, The sum of distances to any point on the ellipse x,y from the two foci c,0 and -c,0 is…

Q: 3.4 Give the set of limit points Ao of a singleton A {(5, 2)} on the plane R2 with the discrete %3D…

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * None…

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

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Q: Suppose X is a topological space, Y is a Hausdorff space, and f: X → Y is continuous. Show that the…

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Q: 1/ Let (X₁T) and (Y₁J.) be top spaces and fix-x be a surjective and continuousmap, then y is Compact…

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Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X,M) := {f €…

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

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Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

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Q: Let X = (1, 2, 3), Tx = (0, X, (1), (2, 3}}, then (X, Tx) is %3D %3D 1) a T1 space 2) a T2 space

A: T1 space: A topological space X has the T1 property if x and y are distinct points of X, there…

Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Example (4): Let (M, d) be a metric space. Define a function e : M X M → R as: e(x,y) d(x,y)…

A: Metric space is the base of topology. A metric space is a function that maps a non-empty set under…

Q: Lemma 2.56 Let (X,T) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f e…

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Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: 5. Prove that the property of a space having an isolated point is a topological property. 6. For any…

A: Sol

Q: Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R 2 with the discrete…

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

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Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Is not closed set in X x X. Then the space X is: (a) Hausdorff Space (b) Not Hausdorff Space (c)…

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Q: Example (3): Let (M, d) be a metric space. Define a function e: Mx M→R by: e(x,y) = Min{1,d(x y)};…

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Q: 5. Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norm of function…

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Q: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

Q: = {a, b, c, d}, t = {Ø, X, {b, c}}. Then in the topological space (X, t), {a} = %3D %3D %3D {a, b}…

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Q: Q/ if (Xi, Txi) and (Yi, Tyi) are topological spaces then if X₁ Yi Prove that X₁ xX₂Y₁xY₂ Where…

A: We will use the basic knowledge of topology and set theory to answer this question correctly and…

Q: If (X.r,) and (X,r,) are T.-spaces, then (X.r,nr,) is also a T,-space. GOOD LUCK

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Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

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Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

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Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

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Q: Wueslien no 0BN Banach space 3. Show that R is a

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Q: Example 1.5. Let X = R? x R² and d1 : X R, such that, (1) di(x, y) = d;((r1, 12), (Y1, Y2)) =| ¤1 –…

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Q: Let X be the subset (R x {0}) U (R x {0}) CR2 Show that the quotient space X/~ is locally Eue line…

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

A: Given - Let X, T and Y, T1 be two topological spaces and let f be a continuous mapping of X into Y.…

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

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Q: Theorem 9.13. Let (X, d) and (Y,e) be metric spaces. Then X × Y is a metric space.

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Q: xamples -X = R with the norm ||x|| = |x| x € R is normed space

A: Here we needs to show real number is a normed space under modules.

Q: Theorem 7.2. LetX,Y be topological spaces and yo E Y. The constant map f : X → Y defined by f(x) =…

A: Given, X, Y be topological spaces and y0∈Y. The constant map f:X→Y defined by fx=y0 Shows in the…

Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: If X is a metric space and Y C X, then Y is a metric space.

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Q: (b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a…

A: X be a normed space and Y be a Banach space We have to show that B(X, Y) - set of all boundes linear…

Q: Let X and Y be two topological sPaces. Jet A and B CX be such that Ā =B. et f:X y be a continuous…

A: Given: X and Y be two topological spaces Let A and B ⊂X be such that A=B

Q: 8- Suppose that X,Y be a normed spaces then the function f: X → Y is continuous at a point xo E X if…

A: True

Q: The graph of f(x,y)=y2-x2 on XYZ space has a saddle point (a,b) if... choice: (choose one correct…

A: Objective is to find the condition for saddle point

Q: X, Y be topological spaces and f : X Y be a continuous map. If X is completely normal then Y is also…

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.