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


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

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