Section 8*: 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
Verify the following version of the principle of recursive definition: Let
be a set. Let
be a function assigning, to every function
mapping a section
of
into
, an element
of
. Then there is a unique function
such that
for each
.
The difference between the two definitions is that the former (in the text) requires
to be defined for any function
that maps a nonempty section of the positive integers to
and also requires
, while the latter (in the exercise) requires
to be additionally specified for the (unique) function
that maps the empty section
to
and also requires the initial value
to be equal to
. As it is easy to see, if, given either
or
, we define the other one by
, then both definitions become equivalent.