Section 51: 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
Given spaces
and
, let
denote the set of homotopy classes of maps of
into
.
(a) Let
. Show that for any
, the set
has a single element.
(b) Show that if
is path connected, then the set
has a single element.
(a) For a continuous
consider
given by
. This is a homotopy between the fixed constant zero map and
.
(b) All “closed” segments of a path are homotopic to the whole path. Since
is path connected, for any two paths we can connect the final point of the first one with the initial point of the second one by a path to form the product of three paths, in which the two given paths are “closed” segments. Given the idea, it can be shown formally, of course.