Section 22*: 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
(a) Let
be a continuous map. Show that if there is a continuous map
such that
equals the identity map of
, then
is a quotient map.
(b) If
, a retraction of
onto
is a continuous map
such that
for each
. Show that a retraction is a quotient map.
(a) We can reformulate as follows: if a continuous function has a continuous right inverse then it is a quotient map. Suppose
has a right inverse, then, by Exercise 5(a) of §2, it is surjective. Since it is also continuous, all we need to show is that it maps open saturated sets to open sets. Let
be open in
. Then,
is open in
as
is continuous.
(b) The inclusion map is a continuous right inverse of the retraction, apply (a).