Section 51: Problem 3 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 space
is said to be contractible if the identity map
is nulhomotopic.
(a) Show that
and
are contractible.
(b) Show that contractible space is path connected.
(c) Show that if
is contractible, then for any
, the set
has a single element.
(d) Show that if
is contractible and
is path connected, then
has a single element.
(a) For either space consider a homotopy
.
(b) Take any two points
and
and a homotopy
such that
and
for some fixed point
. Then
as a function in
defines a path from
to
, and
defines a path from
to
, so that all three points
,
and
are in the same path connected component.
(c) Take a continuous function
and a homotopy
between
and a constant map
. Then
is a homotopy between
and
.
(d) Take a continuous function
and a homotopy
between
and a constant map
. Since
is path connected, there is path
in
from
to a fixed point
. Then
defines a homotopy between
and
.