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) 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 formula_1 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,
formula_2 for formula_3. So "S" is not separable, since any dense subset has to contain at least one point in each formula_4. 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.