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.

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 topic…