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}<t_{2})\,.

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:

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$L$ is a chain.
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$L$ 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 $[\,0,1\,]$ .
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$L$ therefore is not metrizable.
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$L$ is a $1$ –dimensional locally Euclidean
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$L$ therefore is not paracompact.
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$L$ is first countable.
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$L$ is not separable.
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All homotopy groups of $L$ are trivial.
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However, $L$ is not contractible.

Variants

There are several variations of the above construction.

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Instead of $[\,0,\Omega)$ one can use $(0,\Omega)$ or $[\,0,\Omega\,]$ . The latter (obtained by adding a single point to $L$ ) is compact.
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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
Canonical name	LongLine
Date of creation	2013-03-22 13:29:40
Last modified on	2013-03-22 13:29:40
Owner	Dr_Absentius (537)
Last modified by	Dr_Absentius (537)
Numerical id	17
Author	Dr_Absentius (537)
Entry type	Definition
Classification	msc 54G20