Editing Long line (topology) (section)

== Definition ==

The '''closed long ray''' <math>L</math> is defined as the [[Cartesian product]] of the [[First uncountable ordinal|first uncountable ordinal <math>\omega_1</math>]] with the [[Interval (mathematics)|half-open interval]] <math>[0, 1),</math> equipped with the [[order topology]] that arises from the [[lexicographical order]] on <math>\omega_1 \times [0,1)</math>.  The '''open long ray''' is obtained from the closed long ray by removing the smallest element <math>(0, 0).</math>

The '''long line''' is obtained by "gluing" together two long rays, one in the positive direction and the other in the negative 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) (this is the negative half) and the (not reversed) closed long ray (the positive half), 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 <math>\{ 0 \} \times (0, 1)</math> of the one with the same interval of the other but reversing the interval, that is, identify the point <math>(0, t)</math> (where <math>t</math> is a real number such that <math>0 < t < 1</math>) of the one with the point <math>(0, 1 - t)</math> 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''', <math>L^*,</math> is obtained as the [[one-point compactification]] of <math>L</math> by adjoining an additional element to the right end of <math>L.</math>  One can similarly define the '''extended long line''' by adding two elements to the long line, one at each end.