Section 24: 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
Suppose
and
.
is connected (in all three cases). Let
. It is continuous (§21) and
,
. By the Intermediate Value Theorem there is
such that
or
. If
or
the problem is that we cannot take the values at both 0 and 1 and apply the theorem. In fact,
does not have a fixed point if 1 is not in
.