# long line

The long line is a non-paracompact Hausdorff $1$-dimensional manifold constructed as follows. Let $\Omega$ be the first uncountable ordinal (viewed as an ordinal space) and consider the set

 $L:=\Omega\times[0,1)$

endowed with the order topology induced by the lexicographical order, that is the order defined by

 $(\alpha_{1},t_{1})<(\alpha_{2},t_{2})\iff\alpha_{1}<\alpha_{2}\quad\text{or}% \quad(\alpha_{1}=\alpha_{2}\quad\text{and}\quad t_{1}

Intuitively $L$ is obtained by “filling the gaps” between consecutive ordinals in $\Omega$ with intervals, much the same way that nonnegative reals are obtained by filling the gaps between consecutive natural numbers with intervals.

Some of the properties of the long line:

• $L$ is a chain.

• $L$ is not compact; in fact $L$ is not Lindelöf.

Indeed $\left\{\,[\,0,\alpha):\alpha<\Omega\right\}$ is an open cover of $L$ that has no countable subcovering. To see this notice that

 $\bigcup\left\{\,[\,0,\alpha_{x}):x\in X\right\}=\left[\,0,\sup\{\alpha_{x}:x% \in X\}\right)\,$

and since the supremum of a countable collection of countable ordinals is a countable ordinal such a union can never be $[\,0,\Omega)$.

• However, $L$ is sequentially compact.

Indeed every sequence has a convergent subsequence. To see this notice that given a sequence $a:=(a_{n})$ of elements of $L$ there is an ordinal $\alpha$ such that all the terms of $a$ are in the subset $[\,0,\alpha\,]$. Such a subset is compact since it is homeomorphic to $[\,0,1\,]$.

• $L$ therefore is not metrizable.

• $L$ is a $1$–dimensional locally Euclidean

• $L$ therefore is not paracompact.

• $L$ is first countable.

• $L$ is not separable.

• All homotopy groups of $L$ are trivial.

• However, $L$ is not contractible.

## Variants

There are several variations of the above construction.

• Instead of $[\,0,\Omega)$ one can use $(0,\Omega)$ or $[\,0,\Omega\,]$. The latter (obtained by adding a single point to $L$) is compact.

• One can consider the “double” of the above construction. That is the space obtained by gluing two copies of $L$ along $0$. The resulting open manifold is not homeomorphic to $L\setminus\{0\}$.

Title long line LongLine 2013-03-22 13:29:40 2013-03-22 13:29:40 Dr_Absentius (537) Dr_Absentius (537) 17 Dr_Absentius (537) Definition msc 54G20