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long line (Definition)

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

$\displaystyle L:=\Omega \times [0,1)$
endowed with the order topology induced by the lexicographical order, that is the order defined by
$\displaystyle (\alpha _1,t_1) < (\alpha _2,t_2) \iff \alpha _1<\alpha _2$   or$\displaystyle \quad (\alpha _1=\alpha _2$   and$\displaystyle \quad t_1<t_2)\,.$
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:

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\}$.



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"long line" is owned by Dr_Absentius. [ full author list (7) ]
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Cross-references: open, point, variations, contractible, homotopy groups, separable, first countable, paracompact, locally Euclidean, metrizable, homeomorphic, subset, terms, subsequence, convergent, sequence, sequentially compact, union, collection, supremum, countable, open cover, Lindelöf, compact, chain, properties, natural numbers, reals, intervals, consecutive, order, order topology, ordinal space, ordinal, uncountable, manifold, Hausdorff
There are 6 references to this entry.

This is version 14 of long line, born on 2003-02-27, modified 2007-07-18.
Object id is 4069, canonical name is LongLine.
Accessed 4172 times total.

Classification:
AMS MSC54G20 (General topology :: Peculiar spaces :: Counterexamples)

Pending Errata and Addenda
None.
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Please add properties, details, proofs etc by Dr_Absentius on 2003-02-27 04:48:47
This is an entry under construction. The long line has many interesting properties. Feel free to add content.
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