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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Concept explainers

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.

Question

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.

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