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.

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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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