Order topology

On a strictly totally ordered set can be introduced in a natural way a topology that is compatible with the order. This topology is called the order topology. Some topological concepts such as discrete and dense can thus be applied to orders. The concept of order completeness proves in order to not stop Logie for "large" ordered sets as related to the notion of completeness in metric spaces.

Definition

One uses the symbols and to denote elements outside of a strictly totally ordered set X and sets the order " <" to continue by determining sets for all. The open sets of the order topology are:

Other equivalent formulations:

• The order topology on X is the coarsest topology in which the "open intervals" in the sense of the topology are open.
• The "open intervals" form a basis for the order topology.

Note: Some amount of X contains an "infinite " elements, which are denoted by "" and " ". The additional elements from the above definition must be distinguished from the existing X! The introduction of additional elements can be in principle - because of the case distinctions required in evidence at the expense of Argumentationseleganz - avoid by staying on the open intervals in addition itself and sets of the form or enumerating.

Examples

• The real numbers with their usual order " <": The order topology agrees here with the usual topology coincide ( the real numbers as a metric space ).
• The order topology on the set of integers is the discrete topology.
• Sets of ordinals - equipped with its order topology - often used in the topology as counter-examples.

Applications

Due to the order topology can describe some properties of topological orders, there is always a strictly totally ordered set:

• A non-empty, closed, bounded subset contains its infimum and supremum her, if they exist in X.
• The order < is called discrete if it is their order topology. Without topological notions can be characterized as a discrete order:

• A subset S of X is dense in X in the sense of order theory, if between two elements of X is an element s of S is always with. If X is dense in itself in the sense of order theory, then S is dense in X if and only for the purposes of order theory, if S is dense in X with respect to the order topology.
• A discretely ordered set (except in the trivial case of a singleton ) never densely ordered ( in itself ), and vice versa.
• An order is called to order complete if one of the following equivalent conditions holds:

• Each non-empty downward bounded subset has an infimum.
• Every non-empty upwards bounded subset has a supremum. ( The so-called Supremumseigenschaft. )
• Each non-empty bounded set has infimum and supremum.

• Each in itself tight, strict total order ( X, <) can be embedded with the method of " Dedekind cuts " in a proper full order. In the article, Dedekind cut, this is done on the example of rational numbers. This design works well in " large " orders whose order topology can not be metrisieren.

Examples

The slides listed below properties always refer to the quantities in the usual natural order:

Other topologies that are associated with the order

On a strictly totally ordered set (such as the real numbers ) are sometimes called the " half-lines "

As a basis of a topology each, the topology of the limited upward ( 1 ) or the downward limited quantities ( 2 ) is applied. The two topologies are ( on quantities which contain more than one point ) different from each other and the order topology is their least common refinement.

Source

• Boto of Querenburg: set topology. 2nd revised and expanded edition. Springer, Berlin et al 1979, ISBN 3-540-09799-6, p 18 ( high school text).

• Topological structure
• Set topology
• Order theory