In Theorem 10.1 of Topology by Munkres, what is really going on? I need some help to understand it properly...

Question

(1) While proving Theorem 10.1, Munkres talks about non-empty, finite, ordered sets - Does he mean simply ordered set? If so, don't simply ordered set already have an order relation on them which can automatically be used to show them being well-ordered?

(2) Why does he talk about largest elements in the set? Why not smallest?

Its very confusing...

Answer 1

Yes, he is talking about linearly (or simply) ordered sets. In order to show that the linear order is actually a well-order, you’d have to show that every non-empty subset of the set had a least element with respect to the linear order. That can be done in much the same way that Munkres shows that every finite linearly ordered set has a greatest element, but he wants to prove more than just that the set is well-ordered: he wants to prove specifically that it has the order type of a section of $\Bbb Z_+$.

That proof could be done by showing that every finite linearly ordered set has a least element, but the induction step would be just a little messier; he’s chosen the simpler approach. The other version would go as follows, where I follow his language as closely as possible.

Second, we show there is an order-preserving bijection of $A$ with $\{1,\ldots,n\}$ for some $n$. If $A$ has one element, this fact is trivial. Suppose that it is true for sets having $n-1$ elements. Let $b$ be the smallest element of $A$. By hypothesis, there is an order-preserving bijection

$$f':A\setminus\{b\}\to\{1,\ldots,n-1\}\,.$$

Define an order-preserving bijection $f:A\to\{1,\ldots,n\}$ by setting

$$\begin{align*} f(x)&=f'(x)+1,\text{if }x\ne b\\ f(b)&=1\,. \end{align*}$$