An ordinary “line” $\mathbb{R}$ is sometimes thought of as the result of gluing a countable number of copies of a half-open interval $[0, 1)$ 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 $\omega_1$ be the first uncountable ordinal?, and consider $[0, 1)$ as an ordered set. A long ray is the ordered set $\omega_1 \times [0, 1)$ 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 $1$-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.

Every continuous function$f\colon L \to \mathbb{R}$ is eventually constant, i.e., there exists $x \in L$ and $c \in \mathbb{R}$ such that $f(y) = c$ whenever $y \geq x$ (and similarly $f$ is constant for all sufficiently small $x$).

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

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

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