(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (212)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

long line

(Definition)

The long line is a non-paracompact Hausdorff -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

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:

is a chain.
is not compact; in fact is not Lindelöf.
Indeed $\left\{\,[\,0,\alpha ):\alpha <\Omega \right\}$ is an open cover of that has no countable subcovering. To see this notice that

$\displaystyle \bigcup\left\{\, [\,0,\alpha _x):x\in X\right\}=\left[\,0,\sup\{\alpha _x:x\in X\}\right)\,$
and since the supremum of a countable collection of countable ordinals is a countable ordinal such a union can never be $[\,0,\Omega )$ .
However, is sequentially compact.
Indeed every sequence has a convergent subsequence. To see this notice that given a sequence of elements of there is an ordinal $\alpha$ such that all the terms of are in the subset $[\,0,\alpha \,]$ . Such a subset is compact since it is homeomorphic to $[\,0,1\,]$ .
therefore is not metrizable.
is a -dimensional locally Euclidean
therefore is not paracompact.
is first countable.
is not separable.
All homotopy groups of are trivial.
However, is not contractible.

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 ) is compact.
One can consider the “double” of the above construction. That is the space obtained by gluing two copies of along 0. The resulting open manifold is not homeomorphic to $L\setminus \{0\}$ .

Anyone with an account can edit this entry. Please help improve it!

"long line" is owned by Dr_Absentius. [ full author list (7) ]

(view preamble)

Log in to rate this entry.
(view current ratings)

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 MSC:

54G20 (General topology :: Peculiar spaces :: Counterexamples)

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

Interact