In general topology, the **lexicographic ordering on the unit square** 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.

As the name suggests, we use the lexicographical ordering on the square to define a topology. Given two points in the square, say (*x*,*y*) and (*u*,*v*), we say that (*x*,*y*)
≺
(*u*,*v*) if and only if either *x* < *u* or **both** *x* = *u* **and** *y* < *v*. Given the lexicographical ordering on the square, we use the order topology to define the topology on *S*.

The order topology makes *S* into a completely normal Hausdorff space. It is an example of an order topology in which there are uncountably many pairwise-disjoint homeomorphic copies of the real line. Since the lexicographical order on *S* can be proven to be complete, then this topology makes *S* into a compact set. At the same time, *S* is not separable, since the set of all points of the form (x,1/2) is discrete but is uncountable. Hence *S* is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected but not path connected, nor is it locally path connected. Its fundamental group is trivial.