Section 25*: Problem 8 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 component of an open set
. Since
is saturated, it is enough to show that it is open. Since
is locally connected, every component of every its open subset is open, therefore, it is enough to show that if
then
where
is the component of
in
(which is open).
is connected, therefore,
is connected. It follows that
and
.