Question 4. Let f:XY be a mapping from a topological space (X.r) to a topological space (Y.a). Show that the following statements are equivalent: i) f is continuous. i) (A)-7(A) VACx.

Q: Discrete math.  Let S = R. On S we define the relation R as follows (∀x)(∀y)(xRy ⇔ y ≥ x + 3). Is R…

A: The given set S=ℝ. The relation R defined in the set S is ∀x∀yxRy⇔y≥x+3. We have to check whether…

Q: 1.4.10. Prove that the closed interval [0, 1] and the open interval (0,1) have the same cardinality…

A: (Schr¨oder-Bernstein theorem) :Let A and B be sets. Suppose there are injective functions f : A → B…

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: R.2. Let X be the collection of all functions f :R→R. Show that there exists a one-to-one function G…

A: It is given that X is the collection of all functions f: ℝ→ℝ. Show that there exists one-to-one…

Q: 2. Let (X,r) and (Y.A) be two topological spaces. Let B be a base for A. Prove that a function f…

A: Given below the detailed solution

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: Let X be a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: 1.3.2. Let f: X → Y be a function, let B be a subset of Y, and let {B;}ie1 be a family of subsets of…

A: Let f:X→Y be a function, let B be a subset of Y and Bii∈I be a family of subsets of Y.  To prove…

Q: Z are continuous, then their composition Theorem 7.9. If f : X → Y and g gof : X → Z is continuous.…

A: The given theorem is as follows. If f:X→Y,g:Y→Z are continuous, then their composition g∘f:X→Z is…

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: Suppose f X→ Y, and A, B C X. Prove that (a) f(AUB) = ƒ(A) U ƒ(B) (b) ƒ(ANB) ≤ ƒ(A) Ñ ƒ(B)

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Q: Prove: Let X and Y be two topological spaces. Then the function f:X →Y is continuous if and only if…

A: This is a problem of topology.

Q: Let X be a topological space

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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: 0 IS SU b) Let f be a map. from a space X ipto a space Y and 6 is a base for Y .show that f…

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Q: ts) Let f: Z → Z be given by f(x) = 3x + 1. Show that f is one-to-one. Is f onto? EXPLAIN.

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Q: 3. Let X and Y be metric spaces. Let f :X → Y be a function. Definition: If f is discontinuous at a,…

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Q: a) Consider the following functional relation, f, defined as: f:R - R, f(x) = x Determine whether or…

A: (a).                                     Given :  f:ℝ→ℝ  by  f(x)=x2To check: f  is  one - one…

Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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Q: Let ƒ be continuous and nonnegative on [a, b]. Show that there is c € [a, b] with f(c) = (±a få…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

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Q: Theorem 7.21. IfX is normal and f : X → Y is continuous, surjective, and closed, then Y is normal.

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Q: (b) ƒ(AnB) ≤ f(A) Ñ ƒ(B)

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Q: For a bijective function f:(X,r1)→ (Y,12), one of the following statements is true: O f is open if…

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Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: continuous map со

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Q: (e) f is continuous and closed but not open. (f) f is open and closed but not continuous.

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Q: Prove, using the definition of continuity from the topological point of view, that the constant…

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Q: (A) For each metric space 2 and A below determine whether there exists a continuous function f onto…

A: We know  A function(say f:X => Y) is continuous if and only if the  inverse image of every open…

Q: Let f : Z → Z be relation defined by f(x) = ||æ| – 5|+ x for all æ E Z. i) f is a function ii) f is…

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Q: Let f:X Y is a function and let (B: it e A) be an indexed family of Subset of Y. Show that a)(Ua B)…

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

Q: Consider the relation R = {(x,x): x € Z} on Z. Is R reflexive? Symmetric? Transi- tive? If a…

A: We have to check given relation R for reflexive,symmetric and transitive. Concept required: R:A→B is…

Q: 1.) Which of the following are TRUE about differentiable function: I If rx) exists, then the…

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Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

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Q: topological spaces X and Y,

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.