nLab
long line

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Contents

Idea

An ordinary “line” ℝ is sometimes thought of as the result of gluing a countable number of copies of a half-open interval [0,1) end-to-end in both directions. A long line is similarly obtained by gluing an uncountable number of copies of [0,1) 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.

Definition

Definition

Let ω 1 be the first uncountable ordinal?, and consider [0,1) as an ordered set. A long ray is the ordered set ω 1×[0,1) 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 1-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).

Properties

Let L denote the long line, and R the long ray.

  1. Every continuous function f:L→ℝ is eventually constant, i.e., there exists x∈L and c∈ℝ such that f(y)=c whenever y≥x (and similarly f is constant for all sufficiently small x).

  2. L is a normal (T 4) space, but the Tychonoff product L×L¯ with its one-point compactification is not normal. (See for example Munkres.)

  3. Every continuous map L→L has a fixed point.

  4. R is sequentially compact but not compact.

  5. The long line is not contractible. Proof sketch: Suppose H:I×L→L is a homotopy such that H(0,−) is constant and H(1,−) is the identity. For each t∈[0,1] the image imH(t,−) is an interval (either bounded or unbounded), since L is connected. One may show the set

    {t∈I:imH(t,−)is bounded}\{t \in I: \im H(t, -) \text{is bounded}\}

    is both closed and open. It also contains 0, hence is all of I. But it can’t contain t=1, contradiction.

References

  • Wikipedia

  • 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.

Revised on September 8, 2012 19:10:37 by Todd Trimble (67.81.93.25)
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