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 Template:Nowrap and Template:Nowrap Template:Sfnp

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, Template:Nowrap if and only if either Template:Nowrap or both Template:Nowrap and Template:Nowrap. Stated symbolically, $(x, y) ≺ (u, v) ⟺ (x < u) \lor (x = u \land y < v)$

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

The order topology makes S into a completely normal Hausdorff space.Template:Sfnp 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 $0 \leq x \leq 1$ . 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.^[1] Its fundamental group is trivial.Template:Sfnp