Section 21: Problem 2 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Let
and
be metric spaces with metrics
and
, respectively. Let
have the property that for every pair of points
,
of
,
Show that
is an imbedding. It is called an isometric imbedding of
in
.
By the properties of metric it is injective, therefore,
from
onto
is bijective. The image of any open ball in
is open in
and the inverse image of any open ball in
(which, by Exercise 1, is a basis element for the subspace topology of
) is an open ball in
, as
iff
for some
.