Let X be a metric space with metric d.(a) Show that d : X′ × X →  is continuous.(b) Let X′...

Let X be a metric space with metric d.

(a) Show that d : X′ × X →  is continuous.

(b) Let X′ denote a space having the same underlying set as X. Show that if d : X' × X′ →  is continuous, then the topology of X' is finer than the topology of X.

One can summarize the result of this exercise as follows: If X has a metric d, then the topology induced by d is the coarsest topology relative to which the function d is continuous.