Let (Y, d) be a metric space; let X be a space. Define a topology on C(X, Y) as follows: Given ƒ e C(X, Y), and given a positive continuous function 8: X ---» R, on X, let B(ƒ, 8) = {g|d(S(x), g(x)) < 8(x; for all x e X}. (a) Show that the sets B(f, 8) form a basis for a topology on C(X, Y). We call it the fine topology. (b) Show that the fine topology contains the uniform topology. (c) Show that if X is compact, the fine and uniform topologies agree. (d) Show that if X is discrete, then C(X, Y) = Yx and the fine and box topol- ogies agree.

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: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Exercise 8. Any map f: X Y between topological spaces which is continuous at x = X is sequentially…

A: Continuity and sequential continuity are defined below

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: 6.(a) Show that (f.g) = f f(x)g(x)dx satisfies the axioms of an inner product on C[a, b], the space…

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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: 1. Given a set X and a subset A CX, define a function ƒÃ : X → {0, 1} by setting f₁(x) 1 if ¤ ¤ A…

A: (a) fA : X → {0, 1} is defined as             fA(x) = 1 when x ∈ A                       = 0 when x…

Q: Let Id be the identity map in R" and let f : R → R be the function f(x) = ||||². Prove that the…

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Q: * Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((r, y)) = x…

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Q: Let f be continuous over [a,b] and differentiable over (a,b). Then there exists at least one point…

A: Let f be continuous over a,b and differentiable over a,b. Then there exits at leastone point c∈a,b…

Q: 3. Define f(x) so that S(2) = {ap(). f(x) = 0, |피 1 (a) Explain why the function f(x) must be…

A: As per our guidelines we can answer only 3 subparts. please ask others seperately.

Q: Let f : [0, 1] → R be a function with Darboux’s property such that for any y ∈ R, the set  f…

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.