Section 52: Problem 5 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
be a subspace of
; let
. Show that if
is extendable to a continuous map of
into
, then
is the trivial homomorphism (the homomorphism that maps everything to the identity element).
First note, that
is continuous. Then,
is the trivial homomorphism iff for every loop
in
based at
,
iff for every loop
in
based at
,
is nulhomotopic. Consider
defined by
where
is a continuous extension of
. Then,
since
on
and
is a path in
. Moreover,
.