endowed with the order topology induced by the lexicographical order, that is the order defined by
Intuitively is obtained by “filling the gaps” between consecutive ordinals in 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:
There are several variations of the above construction.
Instead of one can use or . 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 . The resulting open manifold is not homeomorphic to .
|Date of creation||2013-03-22 13:29:40|
|Last modified on||2013-03-22 13:29:40|
|Last modified by||Dr_Absentius (537)|