Topology

LIMA, UAEH, 2014

A Space That Is Ordered but Not Metrizable

Well-ordered sets

Definition and examples

  1. Well-orderet set

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

  2. Examples

    • The set with the usual order is well-ordered.
    • The set with the usual order is not well-ordered.

Properties

  1. Properties of well-ordered sets

    • If is well-ordered and has the induced order, then is well-ordered.
    • If are well-ordered sets, then with dictionary order is well-ordered.

Zermelo’s theorem

  1. Zermelo

    If is any set, there is a total order on that is a well-order.

  2. There is a set that is uncountable and well-ordered.

Sections

  1. Section

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

The example

The theorem

  1. There is a set that is uncountable and well-ordered, but every section of it is countable.

  2. Proof

    • Let be an uncountable, well-ordered set. Then the set is also uncountable and well-ordered, 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

  1. If is countable, then has an upper bound in .

  2. 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

  1. A nonmetrizable ordered space

    The set is not metrizable

  2. 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
  3. We will denote with the set .