nLab
long line

Contents

Idea

An ordinary “line\mathbb{R} is sometimes thought of as the result of gluing a countable number of copies of a half-open interval [0,1)[0, 1) end-to-end in both directions. A long line is similarly obtained by gluing an uncountable number of copies of [0,1)[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\omega_1 be the first uncountable ordinal?, and consider [0,1)[0, 1) as an ordered set. A long ray is the ordered set ω 1×[0,1)\omega_1 \times [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 11-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 LL denote the long line, and RR the long ray.

  1. Every continuous function f:Lf\colon L \to \mathbb{R} is eventually constant, i.e., there exists xLx \in L and cc \in \mathbb{R} such that f(y)=cf(y) = c whenever yxy \geq x (and similarly ff is constant for all sufficiently small xx).

  2. LL is a normal (T 4T_4) space, but the Tychonoff product L×L¯L \times \bar{L} with its one-point compactification is not normal. (See for example Munkres.)

  3. Every continuous map LLL \to L has a fixed point.

  4. RR is sequentially compact but not compact.

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

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

    is both closed and open. It also contains 00, hence is all of II. But it can’t contain t=1t = 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)