long line (original) (raw)

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 {[ 0,α):α<Ω} is an open cover of L that has nocountable subcovering. To see this notice that

⋃{[ 0,αx):x∈X}=[ 0,sup⁡{αx:x∈X})

and since the supremum of a countablecollection of countable ordinals is a countable ordinal such a union can never be [ 0,Ω).
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Indeed every sequence has a convergent subsequence. To see this notice that given a sequence a:=(an) of elements of L there is an ordinal α such that all the terms of a are in the subset [ 0,α]. Such a subset is compact since it is homeomorphic to [ 0,1].
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L is a 1–dimensional locally Euclidean
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⋃{[ 0,αx):x∈X}=[ 0,sup⁡{αx:x∈X})
and since the supremum of a countablecollection of countable ordinals is a countable ordinal such a union can never be [ 0,Ω).

Variants

There are several variations of the above construction.

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Instead of [ 0,Ω) one can use (0,Ω) or [ 0,Ω]. 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∖{0}.