An ordinary “line” is sometimes thought of as the result of gluing a countable number of copies of a half-open interval end-to-end in both directions. A long line is similarly obtained by gluing an uncountable number of copies of end-to-end in both directions, and is demonstrably “longer” than an ordinary line on account of a number of peculiar properties.
The long line is a source of many counterexamples in topology.
Let be the first uncountable ordinal?, and consider as an ordered set. A long ray is the ordered set taken in the lexicographic order; as a space, it is given the order topology?. The long line is the space obtained by gluing two long rays together at their initial points.
The long line is a line in the sense of being a -dimensional manifold (without boundary) that is not closed (so not a circle). However, it is not paracompact, so it is not homeomorphic to the real line (even though it is Hausdorff).
Let denote the long line, and the long ray.
Every continuous function is eventually constant, i.e., there exists and such that whenever (and similarly is constant for all sufficiently small ).
Every continuous map has a fixed point.
The long line is not contractible. Proof sketch: Suppose is a homotopy such that is constant and is the identity. For each the image is an interval (either bounded or unbounded), since is connected. One may show the set
is both closed and open. It also contains , hence is all of . But it can’t contain , contradiction.
Steen and Seebach, Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.
J. Munkres, Topology (2nd edition). Prentice-Hall, 2000.