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.
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, for example the intervals
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.
[1] Its fundamental group is trivial.