► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a nonconstant continuous function from X to the discrete space {0, 1}.

Q: C^k is the set of all functions that are continuous and have first k derivatives that are…

A: Ck=set of all functions that are continuous and have first k derivatives that are continuous To show…

Q: if and only if F-1: R(F) →X is bounded.

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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: 5. Consider the function f (X, J) → (Y, U), where (X, T) and (Y, U) are two topological spaces.…

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Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

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Q: 3. Let f: X→ Y be a surjective, continuous map of topological spaces. Prove that if X is compact and…

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Q: Suppose f and g are continuous complex functions on a metric space X. Prove f + g is continuous on X

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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: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Prove that the Peano Curve is continuous and surjective mapping [0,1] onto the unit square

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Q: 2. !. (n!)². > Show that there is no holomorphic function f (z) on any open set containing 0 such…

A: To Prove: There is no holomorphic function fz on any open set containing 0 such that fn0=n!2.

Q: Provide two disjoint nonempty open sets C, D. Is it possible for a function g : R → R to be…

A: A set is a collection of elements like, numbers, objects. The numbers can be called as set of real…

Q: Let f : X → Y be a function where X is a nonempty space and Y is a topological space with a given…

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Q: 20. Let V be an inner product space. Prove that (a) |||x|| - ||y||| ≤ ||x − y||. (b) (x, y) =…

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Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

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Q: {X, Ø, {1}, {2},{1,2},{1,3}}, is (X, t) a connected a) Let X = {1,2,3} with the topology t=…

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Q: Prove that the closed interval [a, b] is homeomorphic to [0, 1]. Use topological definition of…

A: Please to the answer in the next step

Q: (a) Formulate and prove the theorem on boundedness and continuity of linear operators on normed…

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Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: Let fbe a continuous function on an open subset U of a Banach spe into a Banach space Y. Let a and b…

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Q: (b) Let U, T be usual and cofinite topologies on se:t of real numbers R and f be a map from R into R…

A: As per our guidelines we are supposed to answer only first question. So i have solve  question (b)…

Q: Is it possible to have a function on a metric space which is discontimuous at every point of the…

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Q: Exercise 3 Prove that a function f: Rd - Rm is continuous if and only if f¹(U) C Rd is open for any…

A: Given:A function , where  represents the -dimensional Euclidean space and  represents the…

Q: 1. Let X be a topological space. Suppose X contains an infinite, closed, discrete subspace. Show…

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Q: Exercise 2. Let X be a topological space. Endow X X X with the product topology. Consider the map f:…

A: We have to show for every closed subset C , f(C) is closed in X×X

Q: Construct a one-to-on continuous mapping of the set {(#1, 12, - ..., In); 0 < T; < 1. j= 1, 2,…

A: Given that : x1,x2,x3,.........,xn, 0&lt;xj&lt;1, j=1,2,3,.....,n So we have to take ,…

Q: Theorem 8.9. Let f : X Y be a continuous, surjective function. If X is connected, then Y is…

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Q: Let x be non-empty of points. a topological Space having open set with finite numb cal shase wh

A: Let X be a topological space having non-empty open set with finite number of points. Let Y be a…

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: Let X be a topological space with the property that, for every topological space Y, every function f…

A: The space X with the discrete topology means every singleton of space is open.

Q: Exercise 2. Let X be a topological space. Endow Xx X with the product topology. Consider the map f:…

A: We have to give an example of a topological space X such that the diagonal is not closed in X×X

Q: 5.1: The following stat- X is disconnected.

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Q: 10. Let X be a compact space and let Y be a Hausdorff space. Prove that X the projection Ty : X × Y…

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Q: R and C with ||x|| = |x| are Banach spaces.

A: Given: ℝ and ℂ with x=x.  We have to prove that they are Banach spaces.

Q: Let X and Y be Banach spaces and F:X → Y be a linear map which is continuous and open. Will F always…

A: The answer provided below has been developed in a clear step by step manner.