long line

The long line is a non-paracompact Hausdorff $1$-dimensional manifoldconstructed asfollows. Let $\mathrm{\Omega }$ be the first uncountable ordinal (viewed as an ordinal space) and consider the set

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

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

Intuitively $L$ is obtained by “filling the gaps” between consecutiveordinals in $\mathrm{\Omega }$ with intervals, much the same way thatnonnegative reals areobtained 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 is an open cover of $L$ that has nocountable subcovering. To see this notice that

  $$\bigcup \{[\mathrm{\hspace{0.17em}0},{\alpha }_x):x\in X\}=[\mathrm{\hspace{0.17em}0},sup\{{\alpha }_x:x\in X\})$$

  and since the supremum of a countablecollection of countable ordinals is a countable ordinal such a union cannever be $[\mathrm{\hspace{0.17em}0},\mathrm{\Omega })$.

• However, $L$ is sequentially compact.

  Indeed every sequence has a convergent subsequence. To see this notice thatgiven a sequence $a:=(a_n)$ of elements of $L$ there is an ordinal $\alpha$ suchthat all the terms of $a$ are in the subset $[\mathrm{\hspace{0.17em}0},\alpha ]$. Such a subset iscompact since it is homeomorphic to $[\mathrm{\hspace{0.17em}0},1]$.

• $L$ 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 $[\mathrm{\hspace{0.17em}0},\mathrm{\Omega })$ one can use $(0,\mathrm{\Omega })$ or $[\mathrm{\hspace{0.17em}0},\mathrm{\Omega }]$. The latter (obtained by adding a single point to $L$) is compact.

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