# 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:

## 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