## Lexicographic Square, topology

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."

Background:

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, or $a=b$ and $c. 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 $[(a,b),(1,1)).$