# 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.[1] It is an example of an order topology in which there are uncountably many pairwise-disjoint homeomorphic copies of the real line.

## Construction

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) $\scriptstyle\prec$ (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.[1] 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:

$U_{u,v} = \{ (x,y) \in S : (x,y) \prec (u,v) \} \ .$

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.[2]