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 Lindelöf nor separable). 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.

The closed long ray L is defined as the cartesian product 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.

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