Section 25*: Problem 9 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
Following the hint. Let
. We show that
is connected iff
is connected iff
is connected. Indeed, this follows from the fact that
and
are homeomorphisms. Therefore, if
is the component of
then each of
and
is the component of
containing
and
.