nLab
long line

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

The ordinary “line\mathbb{R} (i.e. the real numbers equipped with their Euclidean metric topology) may be thought of as the result of gluing a countable set of copies of the half-open interval [0,1)[0, 1) end-to-end in both directions. A long line is similarly obtained by gluing an uncountable set 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 the half-open interval [0,1)[0, 1) as an totally 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 locally Euclidean space (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 and LL are sequentially compact but not compact. Thus, the image f(L)f(L) under a continuous map f:Lf: L \to \mathbb{R}, being sequentially compact, is a compact subspace of \mathbb{R}.

  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

  • Steen and Seebach, Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995.

  • James Munkres, Topology (2nd edition). Prentice-Hall, 2000.

See also

Revised on May 15, 2017 09:16:13 by Todd Trimble (67.81.95.215)