The long line is a non-paracompact Hausdorff $1$ -dimensional manifold constructed as follows. Let $\GW$ be the first uncountable ordinal (viewed as an ordinal space) and consider the set $$L:=\GW\times [0,1)$$ endowed with the order topology induced by the lexicographical order, that is the order defined by $$(\Ga_1,t_1) < (\Ga_2,t_2) \iff \Ga_1<\Ga_2 \quad\text{or}\quad (\Ga_1=\Ga_2 \quad\text{and}\quad t_1<t_2)\,.$$ Intuitively $L$ is obtained by ``filling the gaps'' between consecutive ordinals in $\GW$ 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 $\left\{\,[\,0,\Ga):\Ga<\GW\right\}$ is an open cover of $L$ that has no countable subcovering. To see this notice that $$\bigcup\left\{\, [\,0,\Ga_x):x\in X\right\}=\left[\,0,\sup\{\Ga_x:x\in X\}\right)\,$$ and since the supremum of a countable collection of countable ordinals is a countable ordinal such a union can never be $[\,0,\GW)$ .

• 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 $\Ga$ such that all the terms of $a$ are in the subset $[\,0,\Ga\,]$ . 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

• Instead of $[\,0,\GW)$ one can use $(0,\GW)$ or $[\,0,\GW\,]$ . 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 $L\setminus \{0\}$ .