Order topology

On a totally ordered set , one can naturally introduce a topology that is compatible with the order. This topology is called the order topology . Some terms from topology and metrics such as discrete , dense and complete can be transferred to orders.

definition

Let a totally ordered set be given.This means that the two laws apply: ${\displaystyle (X,<).}$

${\displaystyle a<b\;\land \;b<c\;\;\Rightarrow \;\;a<c}$	( Transitivity )
either or or ${\displaystyle a<b}$ ${\displaystyle a=b}$ ${\displaystyle b<a}$	( Trichotomy )

for all ${\displaystyle a,b,c\in X.}$

In order to avoid case distinctions at the edge of the interval, the set is first included in the set ${\displaystyle X}$

${\displaystyle {\overline {X}}:=\left\{-\infty \right\}\,\cup _{<}\,X\,\cup _{<}\,\left\{+\infty \right\}}$

embedded and then the intervals are made by means of two limits ${\displaystyle a,b\in {\overline {X}}}$

${\displaystyle \{x\in X\,\mid \,a<x<b\}=:\left]a,b\right[}$

educated. They are all subsets of and define the order topology as a basis in the following way: ${\displaystyle X}$

The open sets of the order topology are the arbitrary, also infinite, union sets of such intervals.

Other, equivalent formulations:

• The order topology on is the coarsest topology in which the (open) intervals are open in the sense of the topology. ${\displaystyle X}$
• The (open) intervals form a basis of the order topology.

What is important is the "strict" property of the order relation, that is, without equality. This makes the intervals (in the manner of speaking of the rational or real numbers) open intervals - as opposed to the closed intervals that contain ${\displaystyle <,}$ ${\displaystyle \leq }$

${\displaystyle \{x\in X\,\mid \,a\leq x\leq b\}\;=:\;\left[a,b\right]}$

and are the complementary sets of open sets. E.g. is

${\displaystyle \left[a,b\right]\;\;=\;\;X\,\setminus \,{\bigl (}\left]-\infty ,b\right[\,\cup \,\left]a,\infty \right[{\bigr )}}$.

If has neither minimum nor maximum, it coincides with the topological closure of in . ${\displaystyle X}$ ${\displaystyle {\overline {X}}}$ ${\displaystyle X}$ ${\displaystyle {\overline {X}}}$

An order topology fulfills the axiom of separation T ₂ and is therefore Hausdorffian .

Applications

The order topology can be used to describe some properties of orders topologically, here is always a strictly total ordered set: ${\displaystyle (X,<)}$

• A non-empty, closed, bounded subset of contains its infimum and its supremum, insofar as they exist in. The latter is always the case if and only if the order is complete . ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle X}$
  

• The order is called discrete if it is its order topology. Without topological terms, a discrete order can be characterized as follows: ${\displaystyle <}$

1. Each element has a unique ancestor, unless it is a minimum of . ${\displaystyle X}$
2. Each element has a unique successor unless it is maximum of . ${\displaystyle X}$

Due to the discrete order, the elements are clearly arranged like strings of pearls, but note the 6th example below.

• A subset of is dense in in the sense of order theory if there is always an element from with between two elements from . Is dense in itself in the sense of order theory, then is dense in in the sense of order theory if and only if dense in is in terms of the order topology. ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle a<b}$ ${\displaystyle X}$ ${\displaystyle s}$ ${\displaystyle S}$ ${\displaystyle a<s<b}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle X}$
• A discretely ordered set is (except in the trivial case of a one-element set) never tightly ordered (in itself) and vice versa.
• Each dense in itself, strict total ordering can be combined with the method of Dedekind cuts in a proper full order to embed. In the article Dedekind's cut , this is shown using the example of rational numbers. This construction also works in orders whose order topology cannot be metrised. ${\displaystyle (X,<)}$ ${\displaystyle ({\hat {X}},<)}$

Examples

The properties mentioned below always relate to the usual, natural order in the quantities:

1. The natural numbers are ordered discretely. Every natural number has a successor.
2. The whole numbers are ordered discretely. Every whole number has a predecessor and a successor.
   The order topology is the discrete.
3. In the case of the real numbers with their usual arrangement , the order topology agrees with the usual topology (the real numbers as metric space ). The real numbers are order-complete. ${\displaystyle <}$
   
4. The rational numbers are not fully ordered, but are closely ordered (in themselves).
5. The rational numbers form a dense subset of the set of real numbers.
6. The set of trunk fractions is arranged discretely. The order clearly consists of two strings of pearls: The order of the negative fractions corresponds to the order of the natural numbers, the order of the positive fractions corresponds to their inversion; is thus order isomorphic to the lexicographically ordered. One of the strings of pearls cannot, however, reach the other through continued predecessor or successor formation. ${\displaystyle B:=\left\{{\tfrac {1}{z}}\,\,\mid \,z\in \mathbb {Z} \setminus \left\{0\right\}\right\}}$ ${\displaystyle B}$   ${\displaystyle {\bigl (}0\!\times \!\mathbb {N} {\bigr )}\,\cup \,{\bigl (}1\!\times \!(-\mathbb {N} ){\bigr )}=:\mathbb {N} \,\cup _{<}\,(-\mathbb {N} ).}$
7. If you add the number 0 from the previous example, the order is no longer discrete, because 0 has neither a predecessor nor a successor. But it is not tight either. ${\displaystyle B}$
   

   The ordinal number . ${\displaystyle \omega +1}$
8. The ordinal number is not ordered discretely: The Limes element has no predecessor, each of its neighborhoods contains an infinite number of natural numbers. (The set of natural numbers is usually referred to as the ordinal number .) ${\displaystyle \omega +1=\mathbb {N} \cup _{<}\{\mathbb {N} \}}$ ${\displaystyle \mathbb {N} =\omega }$ ${\displaystyle \omega }$
9. The order types of and are the same. The latter topology is also the subspace topology induced by, so the analytical convergence in the topological convergence of in . Each countable ordinal can order preserving in embedded. Another example of this kind is that has the same order type as in . ${\displaystyle \omega +1}$ ${\displaystyle \left(1-1/\mathbb {N} _{>0}\right)\cup _{<}\{1\}\;=\;\left\{1-{\tfrac {1}{n}}\,\mid \,n\in \mathbb {N} _{>0}\right\}\cup _{<}\{1\}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \lim _{n\to \infty }\left(1-{\tfrac {1}{n}}\right)=1}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \lim _{n\in \omega }n=\omega }$ ${\displaystyle \omega +1}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \omega ^{2}}$ ${\displaystyle \mathbb {N} -1/\mathbb {N} _{>0}=\left\{k-{\tfrac {1}{n}}\,\mid \,k\in \mathbb {N} \land n\in \mathbb {N} _{>0}\right\}}$ ${\displaystyle \mathbb {Q} }$

Other topologies related to the order

Half straight lines can also be used on a strictly total ordered set

	${\displaystyle \left]a,+\infty \right[:=\{x\in X\,\mid \,a<x\}}$	With ${\displaystyle a\in X}$	(Type A)
or
	${\displaystyle \left]-\infty ,b\right[:=\{x\in X\,\mid \,x<b\}}$	With ${\displaystyle b\in X}$	(Type B)

as the basis of a topology, the topology of the downwardly restricted (type A) or the upwardly restricted sets (type B). The two topologies are - for sets that contain more than one point - different from one another and the order topology is their smallest common refinement . ${\displaystyle X,}$

The concept of convergence in these topologies is very simple: A sequence converges in a topology of type A or B only if it becomes stationary at the corresponding extremum .

References and comments

1. ↑ If the order relation should be given as a weak one, a strict (or strong ) total order is generated from it through the setting ${\displaystyle (X,\leq )}$ ${\displaystyle (X,<)}$

   ${\displaystyle x<y\;\;:\Longleftrightarrow \;\;x\leq y\;\land \;x\neq y.}$

2. ↑ As usual, the following should apply:

   ${\displaystyle -\infty \notin X\not \ni +\infty }$

   and

   ${\displaystyle -\infty <x<+\infty }$for everyone . ${\displaystyle x\in X}$

3. ↑ Exactly the same intervals can be defined without reference to the infinite elements : ${\displaystyle \pm \infty }$

   	${\displaystyle \{x\in X\,\mid \,a<x<b\}}$	${\displaystyle =\left]a,b\right[}$	(limited interval)
   	${\displaystyle \{x\in X\,\mid \,a<x\}}$	${\displaystyle =\left]a,\infty \right[}$	(Interval without right bound in ) ${\displaystyle X}$
   	${\displaystyle \{x\in X\,\mid \,x<b\}}$	${\displaystyle =\left]-\infty ,b\right[}$	(Interval without left bound in ) ${\displaystyle X}$
   	${\displaystyle X}$	${\displaystyle =\left]-\infty ,\infty \right[}$	(the whole room)

   Incidentally, the unrestricted intervals are only needed in the base if there is an extremum on the corresponding side; the whole room even only if it consists of a single element. But if there is, for example, no minimum, then the interval on the left can be unlimited ${\displaystyle X}$ ${\displaystyle X}$
   

   ${\displaystyle \{x\in X\,\mid \,x<b\}=\;\textstyle \bigcup _{a<b}\left]a,b\right[}$

   be formed as a union of base sets.

literature

• Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 18 , doi : 10.1007 / 978-3-642-56860-2 .

See also

• Order type