Long line (topology)
Encyclopedia
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. Therefore it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments[ 0, 1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
of the first uncountable ordinal ω1
with the half-open interval
[ 0, 1), equipped with the order topology
that arises from the lexicographical order
on ω1 ×[ 0, 1). The open long ray is obtained from the closed long ray by removing the smallest element (0,0).
The long line is obtained by putting together a long ray in each direction. More rigorously, it can be defined as the order topology on the disjoint union of the reversed open long ray (“reversed” means the order is reversed) and the (not reversed) closed long ray, totally ordered by letting the points of the latter be greater than the points of the former. Alternatively, take two copies of the open long ray and identify the open interval {0} × (0, 1) of the one with the same interval of the other but reversing the interval, that is, identify the point (0, t) (where t is a real number such that 0 < t < 1) of the one with the point (0,1 − t) of the other, and define the long line to be the topological space obtained by gluing the two open long rays along the open interval identified between the two. (The former construction is better in the sense that it defines the order on the long line and shows that the topology is the order topology; the latter is better in the sense that it uses gluing along an open set, which is clearer from the topological point of view.)
Intuitively, the closed long ray is like a real (closed) half-line, except that it is much longer in one direction: we say that it is long at one end and closed at the other. The open long ray is like the real line (or equivalently an open half-line) except that it is much longer in one direction: we say that it is long at one end and short (open) at the other. The long line is longer than the real lines in both directions: we say that it is long in both directions.
However, many authors speak of the “long line” where we have spoken of the (closed or open) long ray, and there is much confusion between the various long spaces. In many uses or counterexamples, however, the distinction is unessential, because the important part is the “long” end of the line, and it doesn't matter what happens at the other end (whether long, short, or closed).
A related space, the (closed) extended long ray, L*, is obtained as the one-point compactification of L by adjoining an additional element to the right end of L. One can similarly define the extended long line by adding two elements to the long line, one at each end.
[0,1) consists of an uncountable number of copies of [0,1) 'pasted together' end-to-end. Compare this with the fact that for any countable ordinal
α, pasting together α copies of[0,1) gives a space which is still homeomorphic (and order-isomorphic) to [0,1) . (And if we tried to glue together more than ω1 copies of [0,1) , the resulting space would no longer be locally homeomorphic to R.)
Every increasing sequence
in L converges to a limit
in L; this is a consequence of the facts that (1) the elements of ω1 are the countable ordinals, (2) the supremum
of every countable family of countable ordinals is a countable ordinal, and (3) every increasing and bounded sequence of real numbers converges.
Consequently, there can be no strictly increasing function L→R.
As order topologies, the (possibly extended) long rays and lines are normal
Hausdorff space
s. All of them have the same cardinality as the real line, yet they are 'much longer'.
All of them are locally compact. None of them is metrisable
; this can be seen as the long ray is sequentially compact but not compact
, or even Lindelöf
.
The (non-extended) long line or ray is not paracompact. It is path-connected, locally path-connected and simply connected but not contractible. It is a one-dimensional topological manifold
, with boundary in the case of the closed ray. It is first-countable
but not second countable
and not separable, so authors who require the latter properties in their manifolds do not call the long line a manifold.
The long line or ray can be equipped with the structure of a (non-separable) differentiable manifold
(with boundary in the case of the closed ray). However, contrary to the topological structure which is unique (topologically, there is only one way to make the real line "longer" at either end), the differentiable structure is not unique: in fact, for each natural number k there exist infinitely many Ck+1 or C∞ structures on the long line or ray inducing any given Ck structure on it. This is in sharp contrast with the situation for ordinary (that is, separable) manifolds, where a Ck structure uniquely determines a C∞ structure as soon as k≥1.
It makes sense to consider all the long spaces at once because every connected (non-empty) one-dimensional (not necessarily separable) topological manifold
possibly with boundary, is homeomorphic to either the circle, the closed interval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line .
The long line or ray can even be equipped with the structure of a (real) analytic manifold
(with boundary in the case of the closed ray). However, this is much more difficult than for the differentiable case (it depends on the classification of (separable) one-dimensional analytic manifolds, which is more difficult than for differentiable manifolds). Again, any given C∞ structure can be extended in infinitely many ways to different Cω (=analytic) structures.
The long line or ray cannot be equipped with a Riemannian metric that induces its topology.
The reason is that Riemannian manifolds, even without the assumption of paracompactness, can be shown to be metrizable.
The extended long ray L* is compact
. It is the one-point compactification of the closed long ray L, but it is also its Stone-Čech compactification, because any continuous function
from the (closed or open) long ray to the real line is eventually constant. L* is also connected
, but not path-connected
because the long line is 'too long' to be covered by a path, which is a continuous image of an interval. L* is not a manifold and is not first countable.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the long line (or Alexandroff line) is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
somewhat similar to the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments
Definition
The closed long ray L is defined as the cartesian productCartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the first uncountable ordinal ω1
First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals...
with the half-open interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
that arises from the lexicographical order
Lexicographical order
In mathematics, the lexicographic or lexicographical order, , is a generalization of the way the alphabetical order of words is based on the alphabetical order of letters.-Definition:Given two partially ordered sets A and B, the lexicographical order on...
on ω1 ×
The long line is obtained by putting together a long ray in each direction. More rigorously, it can be defined as the order topology on the disjoint union of the reversed open long ray (“reversed” means the order is reversed) and the (not reversed) closed long ray, totally ordered by letting the points of the latter be greater than the points of the former. Alternatively, take two copies of the open long ray and identify the open interval {0} × (0, 1) of the one with the same interval of the other but reversing the interval, that is, identify the point (0, t) (where t is a real number such that 0 < t < 1) of the one with the point (0,1 − t) of the other, and define the long line to be the topological space obtained by gluing the two open long rays along the open interval identified between the two. (The former construction is better in the sense that it defines the order on the long line and shows that the topology is the order topology; the latter is better in the sense that it uses gluing along an open set, which is clearer from the topological point of view.)
Intuitively, the closed long ray is like a real (closed) half-line, except that it is much longer in one direction: we say that it is long at one end and closed at the other. The open long ray is like the real line (or equivalently an open half-line) except that it is much longer in one direction: we say that it is long at one end and short (open) at the other. The long line is longer than the real lines in both directions: we say that it is long in both directions.
However, many authors speak of the “long line” where we have spoken of the (closed or open) long ray, and there is much confusion between the various long spaces. In many uses or counterexamples, however, the distinction is unessential, because the important part is the “long” end of the line, and it doesn't matter what happens at the other end (whether long, short, or closed).
A related space, the (closed) extended long ray, L*, is obtained as the one-point compactification of L by adjoining an additional element to the right end of L. One can similarly define the extended long line by adding two elements to the long line, one at each end.
Properties
The closed long ray L = ω1 ×Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
α, pasting together α copies of
Every increasing sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
in L converges to a limit
Limit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
in L; this is a consequence of the facts that (1) the elements of ω1 are the countable ordinals, (2) the supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
of every countable family of countable ordinals is a countable ordinal, and (3) every increasing and bounded sequence of real numbers converges.
Consequently, there can be no strictly increasing function L→R.
As order topologies, the (possibly extended) long rays and lines are normal
Normal space
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...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s. All of them have the same cardinality as the real line, yet they are 'much longer'.
All of them are locally compact. None of them is metrisable
Metrization theorem
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metricd\colon X \times X \to [0,\infty)...
; this can be seen as the long ray is sequentially compact but not compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, or even Lindelöf
Lindelöf space
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
.
The (non-extended) long line or ray is not paracompact. It is path-connected, locally path-connected and simply connected but not contractible. It is a one-dimensional topological manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, with boundary in the case of the closed ray. It is first-countable
First-countable space
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...
but not second countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
and not separable, so authors who require the latter properties in their manifolds do not call the long line a manifold.
The long line or ray can be equipped with the structure of a (non-separable) differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
(with boundary in the case of the closed ray). However, contrary to the topological structure which is unique (topologically, there is only one way to make the real line "longer" at either end), the differentiable structure is not unique: in fact, for each natural number k there exist infinitely many Ck+1 or C∞ structures on the long line or ray inducing any given Ck structure on it. This is in sharp contrast with the situation for ordinary (that is, separable) manifolds, where a Ck structure uniquely determines a C∞ structure as soon as k≥1.
It makes sense to consider all the long spaces at once because every connected (non-empty) one-dimensional (not necessarily separable) topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
possibly with boundary, is homeomorphic to either the circle, the closed interval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line .
The long line or ray can even be equipped with the structure of a (real) analytic manifold
Analytic manifold
In mathematics, an analytic manifold is a topological manifold with analytic transition maps. Every complex manifold is an analytic manifold....
(with boundary in the case of the closed ray). However, this is much more difficult than for the differentiable case (it depends on the classification of (separable) one-dimensional analytic manifolds, which is more difficult than for differentiable manifolds). Again, any given C∞ structure can be extended in infinitely many ways to different Cω (=analytic) structures.
The long line or ray cannot be equipped with a Riemannian metric that induces its topology.
The reason is that Riemannian manifolds, even without the assumption of paracompactness, can be shown to be metrizable.
The extended long ray L* is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
. It is the one-point compactification of the closed long ray L, but it is also its Stone-Čech compactification, because any continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
from the (closed or open) long ray to the real line is eventually constant. L* is also connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, but not path-connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
because the long line is 'too long' to be covered by a path, which is a continuous image of an interval. L* is not a manifold and is not first countable.