Editing Order topology

{{Short description|Certain topology in mathematics}}
{{Distinguish|Order topology (functional analysis)}}

In [[mathematics]], an '''order topology''' is a specific [[topological space|topology]] that can be defined on any [[totally ordered set]].  It is a natural generalization of the topology of the [[real number]]s to arbitrary totally ordered sets. <!-- why this restriction to totally ordered sets? Partial order relations generate nice non-Hausdorff counter-examples //-->

If ''X'' is a totally ordered set, the '''order topology''' on ''X'' is generated by the [[subbase]] of "open rays"
:<math>\{ x \mid a < x\}</math>
:<math>\{x \mid x < b\}</math>
for all ''a,&nbsp;b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open [[interval (mathematics)|interval]]s
:<math>(a,b) = \{x \mid a < x < b\}</math>
together with the above rays form a [[base (topology)|base]] for the order topology. The [[open set]]s in ''X'' are the sets that are a [[union (set theory)|union]] of (possibly infinitely many) such open intervals and rays.

A [[topological space]] ''X'' is called '''orderable''' or '''linearly orderable'''<ref>{{cite journal |last1=Lynn |first1=I. L. |title=Linearly orderable spaces |journal=[[Proceedings of the American Mathematical Society]] |date=1962 |volume=13 |issue=3 |pages=454–456 |doi=10.1090/S0002-9939-1962-0138089-6 | doi-access=free}}</ref> if there exists a total order on its elements such that the order topology induced by that order and the given topology on ''X'' coincide. The order topology makes ''X'' into a [[completely normal space|completely normal]] [[Hausdorff space]].

The standard topologies on '''R''', '''Q''', '''Z''', and '''N''' are the order topologies.

== Induced order topology ==
If ''Y'' is a subset of ''X'', ''X'' a totally ordered set, then ''Y'' inherits a total order from ''X''. The set ''Y'' therefore has an order topology, the '''induced order topology'''.  As a subset of ''X'', ''Y'' also has a [[subspace topology]]. The subspace topology is always at least as [[finer topology|fine]] as the induced order topology, but they are not in general the same.

For example, consider the subset ''Y'' = {−1} &cup; {1/''n''}<sub>''n''&isin;'''N'''</sub> of the [[rational number|rationals]].  Under the subspace topology, the [[singleton set]] {−1} is open in ''Y'', but under the induced order topology, any open set containing −1 must contain all but finitely many members of the space.

== Example of a subspace of a linearly ordered space whose topology is not an order topology ==
Though the subspace topology of ''Y'' = {−1} &cup; {1/''n''}<sub>''n''&isin;'''N'''</sub> in the section above is shown not to be generated by the induced order on ''Y'', it is nonetheless an order topology on ''Y''; indeed, in the subspace topology every point is isolated (i.e., singleton {''y''} is open in ''Y'' for every ''y'' in ''Y''), so the subspace topology is the [[discrete topology]] on ''Y'' (the topology in which every subset of ''Y'' is open), and the discrete topology on any set is an order topology. To define a total order on ''Y'' that generates the discrete topology on ''Y'', simply modify the induced order on ''Y'' by defining −1 to be the greatest element of ''Y'' and otherwise keeping the same order for the other points, so that in this new order (call it say <<sub>1</sub>) we have 1/''n'' <<sub>1</sub> −1 for all ''n''&nbsp;&isin;&nbsp;'''N'''. Then, in the order topology on ''Y'' generated by <<sub>1</sub>, every point of ''Y'' is isolated in ''Y''.

We wish to define here a subset ''Z'' of a linearly ordered topological space ''X'' such that no total order on ''Z'' generates the subspace topology on ''Z'', so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.

Let <math>Z = \{-1\} \cup (0,1)</math> in the [[real line]]. The same argument as before shows that the subspace topology on ''Z'' is not equal to the induced order topology on ''Z'', but one can show that the subspace topology on ''Z'' cannot be equal to any order topology on ''Z''.

An argument follows. Suppose by way of contradiction that there is some [[Totally ordered set#Strict total order|strict total order]] < on ''Z'' such that the order topology generated by < is equal to the subspace topology on ''Z'' (note that we are not assuming that < is the induced order on ''Z'', but rather an arbitrarily given total order on ''Z'' that generates the subspace topology).

Let ''M''&nbsp;=&nbsp;''Z''&nbsp;\&nbsp;{−1}&nbsp;=&nbsp;(0,1), then ''M'' is [[connected space|connected]], so ''M'' is dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element of ''Z'', then <math>(-\infty,-1)</math> and <math>(-1,\infty)</math> separate ''M'', a contradiction. Assume without loss of generality that −1 is the smallest element of ''Z''. Since {−1} is open in ''Z'', there is some point ''p'' in ''M'' such that the interval (−1,''p'') is [[empty set|empty]], so ''p'' is the minimum of ''M''. Then ''M''&nbsp;\&nbsp;{''p''}&nbsp;=&nbsp;(0,''p'')&nbsp;∪&nbsp;(''p'',1) is not connected with respect to the subspace topology inherited from {{math|'''R'''}}. On the other hand, the subspace topology of ''M''&nbsp;\&nbsp;{''p''} inherited from the order topology of ''Z'' coincides with the order topology of ''M''&nbsp;\&nbsp;{''p''} induced by <, which is connected since there are no gaps in ''M''&nbsp;\&nbsp;{''p''} and it is dense. This is a contradiction.

==Left and right order topologies==
Several variants of the order topology can be given:

* The '''right order topology'''<ref>Steen & Seebach, p. 74</ref> on ''X'' is the topology having as a [[base (topology)|base]] all intervals of the form <math>(a,\infty)=\{x\in X\mid x>a\}</math>, together with the set ''X''.
* The '''left order topology''' on ''X'' is the topology having as a base all intervals of the form <math>(-\infty,a)=\{x\in X\mid x<a\}</math>, together with the set ''X''.

These topologies naturally arise when working with [[Semi-continuity|semicontinuous functions]], in that a real-valued function on a topological space is lower semicontinuous if and only if it is [[Continuous function|continuous]] when the reals are equipped with the right order.<ref name="Stromberg">Stromberg, p. 132, Exercise 4</ref>  The ([[Natural (category theory)|natural]]) [[Compact-open topology|compact open topology]] on the resulting set of continuous functions is sometimes referred to as the ''semicontinuous topology''<ref>{{Cite journal |last=Dal Maso |first=Gianni |date=1993 |title=An Introduction to Γ-Convergence |url=https://link.springer.com/book/10.1007/978-1-4612-0327-8 |journal=SpringerLink |language=en |doi=10.1007/978-1-4612-0327-8|isbn=978-1-4612-6709-6 }}</ref>''.''

Additionally, these topologies can be used to give [[counterexample]]s in general topology.  For example, the left or right order topology on a bounded set provides an example of a [[compact space]] that is not Hausdorff.

The left order topology is the standard topology used for many [[set-theoretic]] purposes on a [[Boolean algebra (structure)|Boolean algebra]].{{Clarify|Boolean algebras are not totally ordered|date=April 2021}}

==Ordinal space==

For any [[ordinal number]] ''&lambda;'' one can consider the spaces of ordinal numbers
:<math>[0,\lambda) = \{\alpha \mid \alpha < \lambda\}</math>
:<math>[0,\lambda] = \{\alpha \mid \alpha \le \lambda\}</math>
together with the natural order topology. These spaces are called '''ordinal spaces'''. (Note that in the usual set-theoretic construction of ordinal numbers we have ''&lambda;'' = [0, ''&lambda;'') and ''&lambda;'' + 1 = [0, ''&lambda;'']). Obviously, these spaces are mostly of interest when ''&lambda;'' is an infinite ordinal; for finite ordinals, the order topology is simply the [[discrete topology]].

When ''&lambda;'' = &omega; (the first infinite ordinal), the space [0,&omega;) is just '''N''' with the usual (still discrete) topology, while [0,&omega;] is the [[Alexandroff_extension|one-point compactification]] of '''N'''.

Of particular interest is the case when ''&lambda;'' = &omega;<sub>1</sub>, the set of all countable ordinals, and the [[first uncountable ordinal]]. The element &omega;<sub>1</sub> is a [[limit point]] of the subset [0,&omega;<sub>1</sub>) even though no [[sequence]] of elements in [0,&omega;<sub>1</sub>) has the element &omega;<sub>1</sub> as its limit. In particular, [0,&omega;<sub>1</sub>] is not [[First-countable space|first-countable]]. The subspace [0,&omega;<sub>1</sub>) is first-countable however, since the only point in [0,&omega;<sub>1</sub>] without a countable [[local base]] is &omega;<sub>1</sub>. Some further properties include
*neither [0,&omega;<sub>1</sub>) or [0,&omega;<sub>1</sub>] is [[separable space|separable]] or [[second-countable]]
*[0,&omega;<sub>1</sub>] is [[compact space|compact]], while [0,&omega;<sub>1</sub>) is [[Sequentially compact space|sequentially compact]] and [[Countably compact space|countably compact]], but not compact or [[paracompact]]

== Topology and ordinals ==<!-- This section is linked from [[Ordinal number]] -->

=== Ordinals as topological spaces ===
Any [[ordinal number]] can be viewed as a topological space by endowing it with the order topology (indeed, ordinals are [[well-order]]ed, so in particular [[totally ordered]]). Unless otherwise specified, this is the usual topology given to ordinals. Moreover, if we are willing to accept a [[proper class]] as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.

The set of [[limit point]]s of an ordinal ''α'' is precisely the set of [[limit ordinal]]s less than ''α''.  [[Successor ordinal]]s (and zero) less than ''α'' are [[isolated point]]s in ''α''.  In particular, the finite ordinals and ω are [[discrete space|discrete]] topological spaces, and no ordinal beyond that is discrete.  The ordinal ''α'' is [[compact space|compact]] as a topological space if and only if ''α'' is either a [[successor ordinal]] or zero.

The [[closed set]]s of a limit ordinal ''α'' are just the closed sets in the sense that we have already defined, namely, those that contain a limit ordinal whenever they contain all sufficiently large ordinals below it.

Any ordinal is, of course, an open subset of any larger ordinal.  We can also define the topology on the ordinals in the following [[recursion|inductive]] way: 0 is the empty topological space, ''α''+1 is obtained by taking the [[Compactification (mathematics)|one-point compactification]] of ''α'', and for ''δ'' a limit ordinal, ''δ'' is equipped with the [[direct limit|inductive limit]] topology. Note that if ''α'' is a successor ordinal, then ''α'' is compact, in which case its one-point compactification ''α''+1 is the [[disjoint union]] of ''α'' and a point.

As topological spaces, all the ordinals are [[Hausdorff space|Hausdorff]] and even [[normal space|normal]].  They are also [[totally disconnected]] (connected components are points), [[scattered space|scattered]] (every non-empty subspace has an isolated point; in this case, just take the smallest element), [[zero-dimensional space|zero-dimensional]] (the topology has a [[clopen]] [[basis (topology)|basis]]: here, write an open interval (''β'',''γ'') as the union of the clopen intervals (''β'',''γ''<nowiki>'</nowiki>+1) = <nowiki>[</nowiki>''β''+1,''γ''<nowiki>']</nowiki> for ''γ''<nowiki>'</nowiki><''γ'').  However, they are not [[extremally disconnected]] in general (there are open sets, for example the even numbers from ω, whose [[closure (mathematics)|closure]] is not open).

The topological spaces ω<sub>1</sub> and its successor ω<sub>1</sub>+1 are frequently used as textbook examples of uncountable topological spaces. For example, in the topological space ω<sub>1</sub>+1, the element ω<sub>1</sub> is in the closure of the subset ω<sub>1</sub> even though no sequence of elements in ω<sub>1</sub> has the element ω<sub>1</sub> as its limit: an element in ω<sub>1</sub> is a countable set; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

The space ω<sub>1</sub> is [[first-countable space|first-countable]] but not [[second-countable space|second-countable]], and ω<sub>1</sub>+1 has neither of these two properties, despite being [[compact space|compact]].  It is also worthy of note that any [[continuous function]] from ω<sub>1</sub> to '''R''' (the [[real line]]) is eventually constant: so the [[Stone–Čech compactification]] of ω<sub>1</sub> is ω<sub>1</sub>+1, just as its one-point compactification (in sharp contrast to ω, whose Stone–Čech compactification is much ''larger'' than ω).

=== Ordinal-indexed sequences ===
If ''α'' is a limit ordinal and ''X'' is a set, an ''α''-indexed sequence of elements of ''X'' merely means a function from ''α'' to ''X''.  This concept, a '''transfinite sequence''' or '''ordinal-indexed sequence''', is a generalization of the concept of a [[sequence]]. An ordinary sequence corresponds to the case ''α'' = ω.

If ''X'' is a topological space, we say that an ''α''-indexed sequence of elements of ''X'' ''converges'' to a limit ''x'' when it converges as a [[net (mathematics)|net]], in other words, when given any [[neighborhood (mathematics)|neighborhood]] ''U'' of ''x'' there is an ordinal ''β'' < ''α'' such that ''x''<sub>''ι''</sub> is in ''U'' for all ''ι'' ≥ ''β''.

Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω<sub>1</sub> is a limit point of ω<sub>1</sub>+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω<sub>1</sub>-indexed sequence which maps any ordinal less than ω<sub>1</sub> to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω<sub>1</sub>, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

However, ordinal-indexed sequences are not powerful enough to replace nets (or [[filter (mathematics)|filter]]s) in general: for example, on the [[Tychonoff plank]] (the product space <math>(\omega_1+1)\times(\omega+1)</math>), the corner point <math>(\omega_1,\omega)</math> is a limit point (it is in the closure) of the open subset <math>\omega_1\times\omega</math>, but it is not the limit of an ordinal-indexed sequence.

== See also ==

* [[List of topologies]]
* [[Lower limit topology]]
* [[Long line (topology)]]
* [[Linear continuum]]
* [[Order topology (functional analysis)]]
* [[Partially ordered space]]

==Notes==
<references/>

==References==
{{refbegin}}
* [[Lynn Arthur Steen|Steen, Lynn A.]] and [[J. Arthur Seebach, Jr.|Seebach, J. Arthur Jr.]]; ''[[Counterexamples in Topology]]'', Holt, Rinehart and Winston (1970). {{isbn|0-03-079485-4}}.
* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
{{refend}}
{{PlanetMath attribution|id=1411|title=Order topology}}

{{Order theory}}

[[Category:General topology]]
[[Category:Order theory]]
[[Category:Ordinal numbers]]
[[Category:Topological spaces]]