long line

The long line is a non-paracompact HausdorffPlanetmathPlanetmath 1-dimensional manifoldMathworldPlanetmath constructed as follows. Let Ω be the first uncountable ordinal (viewed as an ordinal space) and consider the set


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 ordinalsMathworldPlanetmathPlanetmath in Ω with intervals, much the same way that nonnegative reals are obtained by filling the gaps between consecutive natural numbersMathworldPlanetmath with intervals.

Some of the properties of the long line:

  • L is a chain.

  • L is not compactPlanetmathPlanetmath; in fact L is not Lindelöf.

    Indeed {[ 0,α):α<Ω} is an open cover of L that has no countableMathworldPlanetmath subcovering. To see this notice that

    {[ 0,αx):xX}=[ 0,sup{αx:xX})

    and since the supremum of a countable collectionMathworldPlanetmath of countable ordinals is a countable ordinal such a union can never be [ 0,Ω).

  • However, L is sequentially compact.

    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].

  • L therefore is not metrizable.

  • L is a 1–dimensional locally Euclidean

  • L therefore is not paracompact.

  • L is not separablePlanetmathPlanetmath.

  • All homotopy groupsMathworldPlanetmath of L are trivial.

  • However, L is not contractible.


There are several variations of the above construction.

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