Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given topology O (ussually denoted as (Y, O)). Also, let T = {f-'(O) : 0 E O}. Show that T is a topology on X.

Q: 6. (i) Let f: (X, J) → (Y, U), where (X, T) and (Y, U) are two topolog- ical spaces. Show that if…

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Q: Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such…

A: First, we will find the equation of a line given two points.

Q: Show that X'= B(X, K) is a Banach space.

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Q: or not. (c) Let f be a map from a space(X, ). into a space (Y,T) and t,T such that TSt and TS,, show…

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Q: Let f be a function defined from a topological space (X, T) to atopological space (Y,T,).Let a be a…

A: Let f:X,Tx→Y,Ty be a function and a is open in Tx. To prove that the function is continues at a.…

Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

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Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Prove there does not exist a continuous function which maps the closed interval [0, 1] onto the open…

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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: Let f be a function of R sucht hat Ә f(x, y) = f(x,y); for all (x, y) = R then prove that - y^ f(x,…

A: Since given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation for…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: Let 7 and r' be two topologies on a set X and let i : (X,r')(X, T) be the identity map. A) r' is…

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Q: Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1. Show that (φ1,φ2,φ3) form an…

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Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

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Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: (iii) If (X,r) is topological space, A,BCX. Prove that (a) b(A)=AA (b) If ACB, then d(A) ≤d(B)

A: First part Is obvious from definition of boundary points of a set.

Q: Show that H = y:x+2y-z + 1 = 0 is not a sub-space of R³. -1=0} is

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Q: Let (X, τ ) be a discrete space. Is (X, τ ) second countable? Explain why

A: Consider the provided question, Let x,τ be a discrete space,    ⇒x,τ  be denote topology if x,τ is…

Q: 1. Let (X,r) and (Y,A) be two topological spaces. Prove that a function f:(X,r) (Y,A) is continuous…

A: ANSWER :

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: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is F closed?

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Q: Let (Y, d) be a metric space; let X be a space. Define a topology on C(X, Y) as follows: Given ƒ e…

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

Q: lim 2² + y² + z²e¬(z +y° +z²) dædydz = 27. R00 BR

A: Here is the solution of the given problem.Here we use the following co-ordinate transformation to…

Q: Show that d(f₁ g) = sup{ |f(s)-g(s) | : SE5) defines a metric on the space B(S) of Bounded real…

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Q: Let f(x,y,z,w)=45x3z-1/y+wz3. Evaluate f(a,b,c,d) where a,b,c are non-zerol real numbers

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