In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an order topology is a certain
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 ( reflexive) ...
. 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 real ...
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
:
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
** ''Un ...
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 points ...
''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. Th ...
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 to ...
. 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 offe ...
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 rationa ...
. 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
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
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
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 ( reflexi ...
< 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,
, the set of all countable ordinals, and the first uncountable ordinal. The element ω
as its limit. In particular,
. The subspace [0,ω
. Some further properties include
*neither [0,ω
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