Supplementary Exercises*: Well-Ordering: Problem 1 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
Theorem (General principle of recursive definition). Let
be a well-ordered set; let
be a set. Let
be the set of all functions mapping sections of
into
. Given a function
, there exists a unique function
such that
for each
.
[Hint: Follow the pattern outlined in Exercise 10 of §10.]
In Exercise 10 of §10, we are asked to prove a specific case of this theorem for a particular choice of
. Our prove of that result is, in fact, for this more general case. See Exercise 10 of §10.