Order topology

TheInfoList

OR:

In mathematics, an order topology is a certain
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
that can be defined on any
totally ordered set In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
. It is a natural generalization of the topology of the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
to arbitrary totally ordered sets. If ''X'' is a totally ordered set, the order topology on ''X'' is generated by the
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
of "open rays" :$\$ :$\$ for all ''a, b'' in ''X''. Provided ''X'' has at least two elements, this is equivalent to saying that the open intervals :$\left(a,b\right) = \$ together with the above rays form a base for the order topology. The open sets in ''X'' are the sets that are a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
of (possibly infinitely many) such open intervals and rays. A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' is called orderable or linearly orderable 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 In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
. 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 In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
. The subspace topology is always at least as
fine Fine may refer to: Characters * Sylvia Fine (''The Nanny''), Fran's mother on ''The Nanny'' * Officer Fine, a character in ''Tales from the Crypt'', played by Vincent Spano Legal terms * Fine (penalty), money to be paid as punishment for an off ...
as the induced order topology, but they are not in general the same. For example, consider the subset ''Y'' = ∪ ''n''∈N in the
rationals In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
. Under the subspace topology, the singleton set 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.

# An example of a subspace of a linearly ordered space whose topology is not an order topology

Though the subspace topology of ''Y'' = ∪ ''n''∈N in the section above is shown to be not 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 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 an open set), 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 ''<''1) we have 1/''n'' ''<''1 –1 for all ''n'' ∈ N. Then, in the order topology on ''Y'' generated by ''<''1, 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 $Z = \\cup \left(0,1\right)$ 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
strict total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
< 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). In the following, interval notation should be interpreted relative to the < relation. Also, if ''A'' and ''B'' are sets, $A$ shall mean that $a$ for each ''a'' in ''A'' and ''b'' in ''B''. Let ''M'' = ''Z'' \ , the unit interval. ''M'' is connected. If ''m'', ''n'' ∈ ''M'' and ''m'' < -1 < ''n'', then $\left(-\infty, -1\right)$ and $\left(-1, \infty\right)$ separate ''M'', a contradiction. By similar arguments, ''M'' is dense on it self and has no gaps, in regards to <. Thus, ''M'' <  or  < ''M''. Assume without loss of generality that  < ''M''. Since is open in ''Z'', there is some point ''p'' in ''M'' such that the interval (-1, ''p'') is empty. Since  < ''M'', we know -1 is the only element of ''Z'' that is less than ''p'', so ''p'' is the minimum of ''M''. Then ''M'' \  = ''A'' ∪ ''B'', where ''A'' and ''B'' are nonempty open and disjoint subsets of ''M'', given by the intervals of the real line (0,''p'') and (''p'',1) respectively. Notice that the frontier of ''A'' and of ''B'' are both the unitary of ''p''. Assuming without loss of generality ''a'' in ''A'' and ''b'' in ''B'' such that ''a''<''b'', since there are no gaps in ''M'' and it is dense, there is a frontier point between ''A'' and ''B'' in the interval (''a'',''b'') (one can take the supreme of the set of elements ''x'' of ''A'' such that 'a'',''x''is in ''A''). This is a contradiction, since the only frontier is strictly under ''a''.

# Left and right order topologies

Several variants of the order topology can be given: * The right order topologySteen & Seebach, p. 74 on ''X'' is the topology having as a base all intervals of the form $\left(a,\infty\right)=\$, together with the set ''X''. * The left order topology on ''X'' is the topology having as a base all intervals of the form $\left(-\infty,a\right)=\$, together with the set ''X''. The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
that is not Hausdorff. The left order topology is the standard topology used for many
set-theoretic Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
purposes on a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
.

# Ordinal space

For any
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
''λ'' one can consider the spaces of ordinal numbers : :$\left[0,\lambda= \$ 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 ''λ'' = [0,''λ'') and ''λ'' + 1 = [0,''λ'']). Obviously, these spaces are mostly of interest when ''λ'' is an infinite ordinal; otherwise (for finite ordinals), the order topology is simply the discrete topology. When ''λ'' = ω (the first infinite ordinal), the space ,ω)_is_just_N_with_the_usual_(still_discrete)_topology,_while_[0,ωis_the_Alexandroff_extension.html" "title=",ω.html" ;"title=",ω) is just N with the usual (still discrete) topology, while [0,ω">,ω) is just N with the usual (still discrete) topology, while [0,ωis the Alexandroff_extension">one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
of N. Of particular interest is the case when ''λ'' = ω1, the set of all countable ordinals, and the first uncountable ordinal. The element ω1 is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
of the subset [0,ω1) even though no sequence (mathematics), sequence of elements in [0,ω1) has the element ω1 as its limit. In particular, 1is not
first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
. The subspace [0,ω1) is first-countable however, since the only point in 1without a countable local base is ω1. Some further properties include *neither [0,ω1) or 1is separable or
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
* 1is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
, while
sequentially_compact_ In_mathematics,_a_topological_space_''X''_is_sequentially_compact_if_every_sequence_of_points_in_''X''_has_a__convergent_subsequence_converging_to_a_point_in_X. _ Every_metric_space_is_naturally_a_topological_space,_and_for_metric_spaces,_the_notio_...
_and_Countably_compact_space.html" ;"title="Sequentially_compact_space.html" "title=",ω1) is Sequentially compact space">sequentially compact In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notio ...
and Countably compact space">countably compact, but not compact or paracompact

# Topology and ordinals

## Ordinals as topological spaces

Any
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
by endowing it with the order topology (since, being well-ordered, an ordinal is in particular
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
): in the absence of indication to the contrary, it is always that order topology that is meant when an ordinal is thought of as a topological space. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.) The set of
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
s of an ordinal ''α'' is precisely the set of
limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
s less than ''α''. Successor ordinals (and zero) less than ''α'' are
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
as a topological space if and only if ''α'' is a
successor ordinal In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. Properties Every ordinal other than 0 is either a successor ordin ...
. The closed sets 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 further ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, ''α''+1 is obtained by taking the
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Ale ...
of ''α'', and for ''δ'' a limit ordinal, ''δ'' is equipped with the 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 and even normal. They are also
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
(connected components are points), scattered (every non-empty subspace has an isolated point; in this case, just take the smallest element),
zero-dimensional In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical i ...
(the topology has a
clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
: here, write an open interval (''β'',''γ'') as the union of the clopen intervals (''β'',''γ'''+1)= /nowiki>''β''+1,''γ'''/nowiki> for ''γ'''<''γ''). However, they are not extremally disconnected in general (there are open sets, for example the even numbers from ω, whose closure is not open). The topological spaces ω1 and its successor ω1+1 are frequently used as text-book examples of non-countable topological spaces. For example, in the topological space ω1+1, the element ω1 is in the closure of the subset ω1 even though no sequence of elements in ω1 has the element ω1 as its limit: an element in ω1 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 ω1 is
first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
, but not
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
, and ω1+1 has neither of these two properties, despite being
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. It is also worthy of note that any continuous function from ω1 to R (the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
) is eventually constant: so the
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...
of ω1 is ω1+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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
. 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, in other words, when given any neighborhood ''U'' of ''x'' there is an ordinal ''β''<''α'' such that ''x''''ι'' 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, ω1 ( omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number), is a limit point of ω1+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω1-indexed sequence which maps any ordinal less than ω1 to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω1, 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
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component th ...
) in general: for example, on the
Tychonoff plank In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces ,\omega_1/math> and ,\ ...
(the product space $\left(\omega_1+1\right)\times\left(\omega+1\right)$), the corner point $\left(\omega_1,\omega\right)$ is a limit point (it is in the closure) of the open subset $\omega_1\times\omega$, but it is not the limit of an ordinal-indexed sequence.

*
List of topologies The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, ...
*
Lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...
*
Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither L ...
*
Linear continuum In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any ...
* Order topology (functional analysis) * Partially ordered space

# References

* Steen, Lynn A. and Seebach, J. Arthur Jr.; ''
Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) ha ...
'', Holt, Rinehart and Winston (1970). . * Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. * {{Order theory General topology Order theory Ordinal numbers Topological spaces