Lexicographic order topology on the unit square explained

In general topology, the lexicographic ordering on the unit square (sometimes the dictionary order on the unit square^[1]) is a topology on the unit square S, i.e. on the set of points (x,y) in the plane such that and

Construction

The lexicographical ordering gives a total ordering

\prec on the points in the unit square: if (x,y) and (u,v) are two points in the square, if and only if either or both and . Stated symbolically,

The lexicographic order topology on the unit square is the order topology induced by this ordering.

Properties

The order topology makes S into a completely normal Hausdorff space. Since the lexicographical order on S can be proven to be complete, this topology makes S into a compact space. At the same time, S contains an uncountable number of pairwise disjoint open intervals, each homeomorphic to the real line, for example the intervals

U _{x=\{(x,y):1/4<y<1/2\}} for 0 \le x\le 1 . So S is not separable, since any dense subset has to contain at least one point in each U _{x} . Hence S is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected and locally connected, but not path connected and not locally path connected. Its fundamental group is trivial.

See also

• List of topologies
• Long line

Notes and References

1. Book: Introduction to topological manifolds. Lee, John M.. 2011. Springer. 978-1441979391. 2nd. New York. 697506452.