Show that any basic open set about a point on the "top edge," that is, a point of form (a, 1), where a<1, must intersect the "bottom edge."


Definition- The lexicographic square is the set X=[0,1] \times [0,1] with the dictionary, or lexicographic, order. That is (a, b)<(c, d) if and only if either a<b, or a=b and c<d. This is a linear order on X, and the example we seek is X with the order topology.

We follow usual customs for intervals, so that [(a,b),(c,d))=\{   (x,y) \in X : (a,b) \leq (x,y)<(c,d)    \}. A subbase for the order topology on X is the collection of all sets of form [(0,0),(a,b)) or of form [(a,b),(1,1)).