We consider the set C ( [ 0 , 1 ] , R ) of all continuous functions f : [ 0 , 1 ] → R . (i) Show that d 1 : C ( [ 0 , 1 ] , R ) × C ( [ 0 , 1 ] , R ) → R , d 1 ( f , g ) = ∫ 0 1 | f ( x ) − g ( x ) | d x and d 2 : C ( [ 0 , 1 ] , R ) × C ( [ 0 , 1 ] , R ) → R , d 2 ( f , g ) = sup x ∈ [ 0 , 1 ] | f ( x ) − g ( x ) | are metrics on C ( [ 0 , 1 ] , R ) . (ii) Show that the topology T d 2 is finer that the topology T d 1 , i.e., T d 1 ⊂ T d 2 . (iii) Show that the inclusion is strict by finding a sequence of functions which converges only with respect to one of the topologies.
Problem 1 (10 points) (i) Let Y be a topological space. Show that Y is Hausdorff if and only if the diagonal Δ Y = { ( y , y ) ∣ y ∈ Y } ⊂ Y × Y is a closed subset of Y × Y with the Cartesian product topology. (ii) Let X and Y be two topological spaces with Y Hausdorff, and consider two continuous functions f , g : X → Y . Show that Z ⊂ X defined by Z = { x ∈ X ∣ f ( x ) = g ( x ) } is closed in X .