In topology, the long 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 laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
Definition
The closed long rayL is defined as the cartesian product of the first uncountable ordinal ω1 with the half-open interval0, 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 . 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 and the 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 × of the one with the same interval of [the other but reversing the interval, that is, identify the point of the one with the point 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. Intuitively, the closed long ray is like a real 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 except that it is much longer in one direction: we say that it is long at one end and short 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 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. A related space, the 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 × 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 [countableordinal α, pasting together α copies of 0,1) gives [a space which is still homeomorphic . Every increasing sequence in L converges to a limit in L; this is a consequence of the facts that the elements of ω1 are the countable ordinals, the supremum of every countable family of countable ordinals is a countable ordinal, and every increasing and bounded sequence of real numbers converges. Consequently, there can be no strictly increasing functionL→R. In fact, every continuous functionL→R is eventually constant. As order topologies, the long rays and lines are normalHausdorff spaces. 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 metrizable; this can be seen as the long ray is sequentially compact but not compact, or even Lindelöf. The 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. It makes sense to consider all the long spaces at once because every connected one-dimensional topological manifold possibly with boundary, is homeomorphic to either the circle, the closed interval, the open interval, the half-open interval, the closed long ray, the open long ray, or the long line. The long line or ray can be equipped with the structure of a differentiable manifold. However, contrary to the topological structure which is unique, the differentiable structure is not unique: in fact, there are uncountably many pairwise non-diffeomorphic smooth structures on it. This is in sharp contrast to the real line, where there are also different smooth structures, but all of them are diffeomorphic to the standard one. The long line or ray can even be equipped with the structure of a analytic manifold. However, this is much more difficult than for the differentiable case. Again, any given C∞ structure can be extended in infinitely many ways to different Cω 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 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.
There exists a p-adic analog of the long line, which is due to George Bergman. This space is constructed as the increasing union of an uncountable directed set of copies Xγ of the ring of p-adic integers, indexed by a countable ordinal γ. Define a map from Xδ to Xγ whenever δ<γ as follows:
If γ is a successor ε+1 then the map from Xε to Xγ is just multiplication by p. For other δ the map from Xδ to Xγ is the composition of the map from Xδ to Xε and the map from Xε to Xγ
If γ is a limit ordinal then the direct limit of the sets Xδ for δ<γ is a countable union of p-adic balls, so can be embedded in Xγ, as Xγ with a point removed is also a countable union of p-adic balls. This defines compatible embeddings of Xδ into Xγ for all δ<γ.
This space is not compact, but the union of any countable set of compact subspaces has compact closure.
Higher dimensions
Some examples of non-paracompact manifolds in higher dimensions include the Prüfer manifold, products of any non-paracompact manifold with any non-empty manifold, the ball of long radius, and so on. The bagpipe theorem shows that there are 2ℵ1 isomorphism classes of non-paracompact surfaces. There are no complex analogues of the long line as every Riemann surface is paracompact, but gave an example of a non-paracompact complex manifold of complex dimension 2.
