Section 33: Problem 5 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 such function exists. For
and
, and for
and
, apply the previous exercise. Suppose now we have a pair of disjoint closed
sets. Using the previous exercise we construct
such that
and
. Then
is what we need. ( Compare to exercise 3: for any closed
,
satisfies all the properties of a function constructed in exercise 4. )