Wellordered sets
Definition and examples

Wellorderet set
We say that the totally ordered set is wellordered if every nonempty subset has a minimum element.

Examples
 The set with the usual order is wellordered.
 The set with the usual order is not wellordered.
Properties

Properties of wellordered sets
 If is wellordered and has the induced order, then is wellordered.
 If are wellordered sets, then with dictionary order is wellordered.
Zermelo’s theorem

Zermelo
If is any set, there is a total order on that is a wellorder.

There is a set that is uncountable and wellordered.
Sections

Section
If is an ordered set and , we denote by the set: is called the section of by .
The example
The theorem

There is a set that is uncountable and wellordered, but every section of it is countable.

Proof
 Let be an uncountable, wellordered set. Then the set is also uncountable and wellordered, and has sections that are uncountable (for example, those of elements of the form .)
 The set S_{\alpha} is nonempty, let its smallest element.
 Then is the required set.
A corollary

If is countable, then has an upper bound in .

Proof
 Since for each , we have that is countable, then is also countable.
 Since , and is uncountable, we can choose .
 If there was with , then would be in , which is a contradiction.
 Hence is an upper bound of .\qed
A nonmetrizable ordered space

A nonmetrizable ordered space
The set is not metrizable

Proof
 We have that is a limit point of , since any basic element containing has the form and must intersect . Otherwise we would have , but is countable.
 However, there is no sequence in that converges to , since if is a sequence in , there is an upper bound for all the terms in the sequence. Then contains no point of the sequence.\qed

We will denote with the set .