Lexicographic order topology on the unit square
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. It is an example of an order topology in which there are uncountably many pairwise-disjoint homeomorphic copies of the real line.
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. For each point (u,v) we get an open set, denoted Uu,v, given by all the points in S that precede (u,v) with respect to the lexicographical ordering:
The open sets in the lexicographic ordering on the unit square are the whole set S, the empty set ∅, and those sets generated by the Uu,v, i.e. the sets formed by all possible unions of finite intersections.