# Lexicographic order topology on the unit square

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 0 ≤ *x* ≤ 1 and 0 ≤ *y* ≤ 1.^{[2]}

## Construction[]

The lexicographical ordering gives a total ordering on the points in the unit square: if (*x*,*y*) and (*u*,*v*) are two points in the square, (*x*,*y*) (*u*,*v*) if and only if either *x* < *u* or **both** *x* = *u* **and** *y* < *v*. Stated symbolically,

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

## Properties[]

The order topology makes *S* into a completely normal Hausdorff space.^{[3]} 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, namely the intervals for . So *S* is not separable, since any dense subset has to contain at least one point in each . 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.^{[1]} Its fundamental group is trivial.^{[2]}

## See also[]

- List of topologies
- Long line

## Notes[]

- ^
^{a}^{b}1950-, Lee, John M. (2011).*Introduction to topological manifolds*(2nd ed.). New York: Springer. ISBN 978-1441979391. OCLC 697506452.CS1 maint: numeric names: authors list (link) - ^
^{a}^{b}Steen & Seebach (1995), p. 73. **^**Steen & Seebach (1995), p. 66.

## References[]

- Steen, L. A.; Seebach, J. A. (1995),
*Counterexamples in Topology*, Dover, ISBN 0-486-68735-X

- General topology
- Topological spaces