long line
The long line is a nonparacompact Hausdorff^{} $1$dimensional manifold^{} constructed as follows. 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 the lexicographical order, that is the order defined by
$$ 
Intuitively $L$ is obtained by “filling the gaps” between consecutive ordinals^{} in $\mathrm{\Omega}$ 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 $$ is an open cover of $L$ that has no countable^{} subcovering. To see this notice that
$$\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 countable collection^{} of countable ordinals is a countable ordinal such a union can never be $[\mathrm{\hspace{0.17em}0},\mathrm{\Omega})$.

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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 $[\mathrm{\hspace{0.17em}0},\alpha ]$. Such a subset is compact since it is homeomorphic to $[\mathrm{\hspace{0.17em}0},1]$.

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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 $[\mathrm{\hspace{0.17em}0},\mathrm{\Omega})$ 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.

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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  20130322 13:29:40 
Last modified on  20130322 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 