► Exercise 4.4. Prove that a topological space X is disconnected if and only there exists a nonconstant continuous function from X to the discrete space {0, 1}.

Prove that the closed interval [a, b] is homeomorphic to [0, 1]. Use topological definition of continuity using open sets (NOT a delta epsilon proof) and be specific.

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.

Show that d(f₁g) = sup { | f(s)-g(s)/: se}} defines metric on the space B(S) a of Bounded real valued functions on a sut S.