Section 21: Problem 1 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
. If
is a metric for the topology of
, show that
is a metric for the subspace topology on
.
The collection of balls
centered at points
in the metric
is a basis for the metric topology on
.
as a subspace has a basis element
for all
. Clearly,
so that the subspace topology on
is finer than the metric topology on
. Further, consider any basis element
for the subspace topology. If
, let
. Then
. Therefore,
is open in the metric topology on
, and the metric topology on
is finer than the subspace topology on
.