Question

Let X be a metric space with metric d. (a) Show that d : X × X ? R is continuous. (b) Let X' denote a space having the same underlying set as X. Show that if d : X' × X' ? R 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.

          Let X be a metric space with metric d.
(a) Show that d : X × X ? R is continuous.
(b) Let X' denote a space having the same underlying set as X. Show that if d : X' × X' ? R 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.

Let X be a metric space with metric d.
(a) Show that d : X × X ? R is continuous.
(b) Let X' denote a space having the same underlying set as X. Show that if d : X' × X' ? R 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.

Added by Carla C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

12/13/2022

Step 1

This means that for any open set U in R, the preimage of U under d is an open set in X' x X. Let U be an open set in R. Then for any point (x,y) in the preimage of U, there exists an ε > 0 such that the open ball B(d(x,y), ε) is contained in U. Now consider Show more…

Show all steps

Thanks for your feedback!

Let X be a metric space with metric d. (a) Show that d : X x X → R is continuous. (b) Let X' denote a space having the same underlying set as X. Show that if d : X' x X' → R 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.